This has been CNBC's live blog covering updates on the war in Ukraine. [Follow the latest updates here.]

Remaining Ukrainian forces in the southern port of Mariupol continue to fight, Ukrainian Prime Minister Denys Shmyhal said on Sunday, defying a Russian demand that they surrender.

Moscow said on Saturday its troops had seized the urban area of Mariupol. Only a small contingent of Ukrainian fighters remained inside a steelworks complex in the besieged southern port, according to reports by Reuters.

Meanwhile, at least five people were dead and 20 were injured in Kharkiv after a Russian missile attack on Ukraine's second biggest city, according to officials.

Russia also said it had destroyed production buildings of an armored vehicle plant in Kyiv and a military repair facility in the city of Mykolaiv, the Interfax news agency quoted Russia's defense ministry as saying on Saturday.

Ukrainian arrivals at U.S.-Mexico border surge

Ukrainians now rank as the third most dominant nationality in encounters at the U.S.-Mexico border, according to internal documents obtained by NBC News.

Over the past week, authorities encountered an average of 1,058 Ukrainians a day, NBC reported.

This has contributed to the highest number of encounters recorded at the border since 2000, with 221,303 encounters recorded in March. That's roughly 8,000 more than previous high reached in July.

Ukrainians are allowed to claim asylum in the U.S. following Russia's invasion.

— Chelsea Ong

Austrian chancellor: Putin believes he is winning war with Ukraine

Austrian Chancellor Karl Nehammer met in person with Vladimir Putin in Moscow this week and said the Russian president believes he is winning the war with Ukraine. Nehammer is the first European leader to do so since the invasion began on Feb. 24.

In an interview on NBC's "Meet the Press," Nehammer said, "We have to look in his eyes and we have to confront him with ... what we see in Ukraine.''

Nehammer said his discussion with Putin "was not a friendly conversation. It was a frank and tough conversation."

"I told him what I saw. I saw the war crimes. I saw the massive loss of the Russian army," he said.

Nehammer added, however, that while Putin believes he is winning the war, the Russian president told him: "It's better the war ends earlier than later. So I think he knows exactly what's going on now."

— Terri Cullen

Disclosure: NBCUniversal is parent of NBC and CNBC.

Mariupol 'doesn't exist anymore' after widespread destruction, foreign minister says

Ukrainian Foreign Minister Dmytro Kuleba described an increasingly dire situation in Mariupol, saying that the key port city effectively "doesn't exist anymore."

The southeastern city has been under relentless bombing from Russian forces. Roughly 95% of the city has been destroyed in bombing, the Ukrainian government said on its official Twitter account. Mayor Vadym Boychenko said in an interview with the Associated Press last week that at least 21,000 people were killed, while bodies were "carpeted through the streets." 

It's challenging to determine the extent of the deaths in Mariupol, as the city is effectively cut off from communication. An estimated 120,000 people remain in the city, Boychenko added, a far cry from its pre-war population of 450,000. Many Ukrainians who remain trapped are left with dwindling food supplies, water, a lack of medicine, electricity or heat.

"The situation in Mariupol is both dire militarily and heartbreaking," Kuleba said in an interview with CBS' "Face the Nation." "The city doesn't exist anymore," the foreign minister added, saying what's left of "the Ukrainian army and large group of civilians are basically encircled by the Russian forces."

Ukraine and international aid organizations have struggled to evacuate civilians. Ukrainian and Russian officials have been unable to agree on conditions to open a humanitarian corridor. The latest was supposed to be opened Sunday.

"We are working hard to get the humanitarian corridors back on track as soon as possible," Ukrainian Deputy Prime Minister Iryna Vereshchuk said in a translated Telegram post.

—Jessica Bursztynsky

At least 202 children killed in Ukraine since the start of war, prosecutors say

At least 202 children have been killed since Russia began its invasion of Ukraine, according to the Ukrainian Prosecutor General's office. More than 361 children have been injured in the war, the office said, citing data from juvenile prosecutors.

Officials have stressed the figures are not final, as it's difficult to confirm reports in places with heavy fighting or that are occupied by Russian forces. Russia has repeatedly denied targeting civilians, despite evidence to the contrary.

In addition to the casualties, the prosecutors said Russian troops have damaged 1,018 educational institutions. Of those, 95 were completely destroyed.

— Jessica Bursztynsky

Zelenskyy says he spoke with IMF director to discuss financial stability, post-war plans

Ukrainian President Volodymyr Zelenskyy said he's spoken with International Monetary Fund Director Kristalina Georgieva about the country's finances and post-war plans.

"Discussed with IMF Managing Director @KGeorgieva the issue of ensuring Ukraine's financial stability & preparations for post-war reconstruction," Zelenskyy said on Twitter. "We have clear plans for now, as well as a vision of prospects. I'm sure cooperation between the IMF & [Ukraine] will continue to be fruitful."

Russia's invasion of Ukraine, which began in late February, has caused significant economic damage in the country. Earlier Sunday, an economic advisor to Zelenskyy suggested Ukraine may need a $1 trillion to cover its losses from the war.

— Alex Sherman

Ukrainians identify dead relatives as volunteers dig up civilians killed by Russian soldiers

Volunteers in Mykulychi, Ukraine, a village outside the capital city of Kyiv, are digging up bodies of civilians killed by Russian soldiers that were temporarily buried by local residents, according to the Associated Press.

More than 900 civilian bodies have been found in towns and villages around Kyiv. Volunteers are digging up bodies placed in temporary caskets weeks ago during a monthlong Russian occupation. Loved ones have recently returned to their hometowns as violence has subsided, the AP reported.

Wives of men shot and killed by Russian soldiers have identified their husbands while grieving their losses together. Some bodies that have been dug up have yet to be identified by relatives, the report said.

Two neighbors, Ira Slepchenko and Valya Naumenko, returned to identify their husbands, knowing where they'd been temporarily buried. According to the AP, on the final day of the Russian occupation, Russian soldiers knocked on doors of local residents. The AP reported that when Naumenko's husband, Pavlo Ivanyuk, opened the door, soldiers took him to the house's garage and shot him in the head.

Slepchenko's husband, Sasha Nedolezhko, heard the shot, the report said. He opened his door and the soldiers shot him, too.

"I want this war to end as soon as possible," Slepchenko told the AP.

—Alex Sherman

At least five dead, 20 injured in attack on Kharkiv

Five people were killed in the Russian shelling of Kharkiv, regional governor Oleh Syniehubov said on his Telegram channel. At least 20 people were injured, he added.

Syniehubov said the center of Kharkiv came under shelling by multiple rocket launchers, damaging residential buildings and city infrastructure. Kharkiv, Ukraine's second biggest city, has been faced with intermittent bombing since Russia began its full scale invasion in late February.

—Jessica Bursztynsky

World Central Kitchen sets up new location after missile strike

Chef Jose Andres's nonprofit World Central Kitchen is resuming operations in the Ukrainian city of Kharkiv a day after a missile strike destroyed its restaurant.

Four staff were wounded in the attack, but are doing well, the agency's CEO, Nate Mook, said on Twitter. "Today, the restaurant team is moving all food products & non-damaged equipment to another kitchen location in Kharkiv," Mook said.

Andres added that "everyone is ready and willing to start cooking in another location."

—Jessica Bursztynsky

U.S. President Biden, on Easter, prays for peace

U.S. President Joe Biden says he's praying on Easter for those living in the "dark shadow" of war, persecution and poverty.

Biden released an Easter message Sunday in which he says he's also praying for peace, freedom and basic dignity and respect for all of God's children.

Biden didn't say which war he had in mind, but the president has been deeply involved in trying to force an end to Russia's invasion of Ukraine.

The American president says he's grateful that the easing of the Covid-19 pandemic has allow many people around the world to celebrate by attending religious services and in-person family gatherings. He also acknowledges that the holiest day on the Christian calendar "falls on heavy hearts for those who have lost loved ones and those among us living in the dark shadow of war, persecution and poverty."

— Associated Press

Ukraine is running a monthly deficit of $5 billion during the war, PM says

Ukrainian Prime Minister Denys Shmyhal said his country is running a budget deficit of about $5 billion a month during the war.

"We need more money for executing our humanitarian and social obligations," Shmyhal said on ABC's "This Week." "Now only half of our economy is working. So we ask for financial support... We appreciate and we are so grateful for the financial support from the United States and all of our international partners."

This week, U.S. President Joe Biden announced another $800 million in military assistance following a phone call with Ukrainian President Volodymyr Zelenskyy. Last month, Biden signed a $1.5 trillion bill that includes $13.6 billion for assistance to Ukraine, which fits into a broader U.S. effort to bolster Ukrainian defense, hamper Russia's economy and support civilians displaced by the war.

Shmyhal said Ukraine's finance team will attend the spring meetings of the International Monetary Fund and the World Bank in Washington.

— Yun Li

Two killed by Russian shelling in Zolote, regional governor says

Russian forces opened fire on civilians in the eastern Ukrainian town of Zolote, killing at least two and injuring four, the governor of Luhansk said.

"This is a premeditated murder," Serhiy Haidai said in a statement posted to his Telegram channel. "There is nothing nearby except residential buildings. The Russians purposefully hit the population."

CNBC could not immediately verify the claims. Russia has denied it is targeting civilians in its invasion of Ukraine, despite well-documented evidence to the contrary.

— Jessica Bursztynsky

Ukrainian prime minister says forces in Mariupol have not surrendered

Remaining Ukrainian forces in the southern port of Mariupol are still fighting and continue to defy a Russian demand that they surrender, Ukrainian Prime Minister Denys Shmyhal said on Sunday.

"The city still has not fallen," Shmyhal told ABC's "This Week" program, adding that Ukrainian soldiers continue to control some parts of the city.

Shmyhal said that he will attend the International Monetary Fund and World Bank meetings in Washington this week and will seek more financial assistance for Ukraine.

— Reuters

Zelenskyy expects Biden to visit Ukraine

Ukrainian leader Volodymyr Zelenskyy said he wants U.S. President Joe Biden to visit the embattled country, in an interview on CNN's "State of the Union."

"I think he will," Zelenskyy said. "It's his decision of course and about the safety situation, it depends. But I think he's the leader of the United States and that's why he should come here to see."

The White House has said previously there are no plans for Biden to travel to Ukraine, but some have been calling on Biden to visit after a handful of leaders visited the capital of Kyiv.

— Jessica Bursztynsky

Zelenskyy says world should prepare for Putin to use nuclear weapons

Ukrainian President Volodymyr Zelenskyy said global leaders should prepare for Russian President Vladimir Putin to use nuclear weapons in his war on Ukraine.

"I think all of the world, all of the countries have to be worried," Zelenskyy said in an interview that aired on CNN's "State of the Union."

"For them, life of the people is nothing," Zelenskyy said of the Kremlin.

—Jessica Bursztynsky

Pope laments 'Easter of war' in address to crowds in Rome

Pope Francis has lamented what he described as an "Easter of war" in his 'Urbi et Orbi' message on Sunday, and again called for peace in Ukraine.

Addressing around 50,000 pilgrims and followers in St. Peter's Square, Pope Francis dedicated much of his blessing to Ukraine and implicitly criticized Russia for what he called a "cruel and senseless" conflict.

"Our eyes, too, are incredulous on this Easter of war. We have seen all too much blood, all too much violence. Our hearts, too, have been filled with fear and anguish, as so many of our brothers and sisters have had to lock themselves away in order to be safe from bombing," he said.

Ukraine, he said, was "sorely tried by the violence and destruction of the cruel and senseless war into which it was dragged."

— Holly Ellyatt

Ukraine needs $1 trillion to cover losses from the war, senior official says

Ukraine will need a trillion dollars to cover its losses from the war with Russia, the president's economic adviser Oleh Ustenko said on Sunday.

Speaking on national television, Ustenko also said Ukraine has asked G-7 nations for $50 billion in financial support and is also considering issuing 0% coupon bonds to help it cover a war-linked budget deficit over the next six months, Reuters reported.

Ustenko said these options were being actively discussed.

— Holly Ellyatt

Russia says it has hit an ammunition factory near Kyiv

Russia has claimed that its armed forces have destroyed an ammunition factory near Kyiv.

"During the night, high-precision air-based missiles destroyed an ammunition factory near Brovary in Kiev region," Russia's Defence Ministry said in a briefing on the messenger app Telegram on Sunday.

Earlier today there were reports of explosions in Ukraine's capital Kyiv but there had been little information regarding the target.

NBC News has not independently verified their claims.

— Holly Ellyatt

Ukrainian fighters in Mariupol told 'their lives will be spared' if they surrender

Russia says it has offered Ukrainian fighters left in Mariupol, the besieged southern port city that Russia claimed to have seized on Saturday, a last remaining window to surrender.

Russia said in a statement reported by state news agency Tass that it was giving Ukrainian "militants" and "foreign mercenaries" left in Mariupol a chance to lay down their arms from 6:00 a.m. Moscow time on Sunday (3:00 a.m. London time) and that if they did so "their lives will be spared."

The statement was aimed at the remaining Ukrainian forces believed to be holding out in the massive Azovstal iron and steel works complex. Ukrainian forces have spent weeks resisting Russia's takeover of the strategically-important port city Mariupol, which has faced relentless bombardment and now lies largely in ruins.

Russia said that if the offer was accepted, Ukrainian fighters should raise white flags around the perimeter of the steelworks.

There has been no indication that Ukrainian fighters plan to surrender, and the deadline has now passed.

— Holly Ellyatt

Russia continues to shift troops, bombard eastern Ukraine

Moscow is still redeploying forces to the eastern part of Ukraine and attacking defenders there with artillery as it gets ready to restart its offensive in that part of the country, the U.K. Ministry of Defence said on Sunday.

Russian combat and support units are shifting to Kharkiv as well as to Severdonetsk, in Ukraine's Luhansk province, among other locations, the ministry said.

Kharkiv, Ukraine's second biggest city, has been under intermittent bombardment since the war began in late February.

The redeployments include troops that had withdrawn to Belarus, the British ministry said.

"Though Russia's operational focus has shifted to eastern Ukraine, Russia's ultimate objective remains the same," the ministry said on Twitter. "It is committed to compelling Ukraine to abandon its Euro-Atlantic orientation and asserting its own regional dominance."

— Ted Kemp

Zelenskyy says Mariupol's fate is key to peace talks

Ukrainian President Volodymyr Zelenskyy told Ukrainian media that the continuing siege of Mariupol could scuttle any attempts to find a negotiated end to the war.

"The destruction of all our guys in Mariupol – what they are doing now – can put an end to any format of negotiations," Zelenskyy said in an interview.

Zelenskyy also said he spoke with the leaders of Britain and Sweden about how best to help those defending Mariupol and the tens of thousands of civilians trapped there.

Mariupol's fate can be decided either through battle or diplomacy, he said.

"Either our partners give Ukraine all of the necessary heavy weapons, the planes, and without exaggeration immediately, so we can reduce the pressure of the occupiers on Mariupol and break the blockade," he said in his nightly video address to the nation. "Or we do so through negotiations, in which the role of our partners should be decisive."

Zelenskyy said the situation in Mariupol remains "inhuman" and that Russia "is deliberately trying to destroy everyone who is there."

The president's office said the southern port city is holding out but the situation is critical. The battle for Mariupol has come at a horrific cost to trapped and starving civilians. Locals reported seeing Russian troops digging up bodies from residential courtyards and prohibiting new burials. It was unclear why.

Russian Defense Ministry spokesman Igor Konashenkov said Saturday that Ukrainian forces have been driven out of most of the city and remain only in the huge Azovstal steel mill.

Mariupol's capture would allow Russian forces from the Crimean Peninsula to fully link up with troops in the Donbas region, Ukraine's eastern industrial heartland.

— Associated Press

Russia says its forces take most of Mariupol

Russia says it has seized the urban area of Mariupol and only a small contingent of Ukrainian fighters remain inside the steelworks in the besieged southern port, according to a report from Reuters. It would be the first major city to have fallen to Russian forces since the Feb. 24 invasion.

"The situation is very difficult" in Mariupol, President Volodymyr Zelenskyy told the Ukrainska Pravda news portal. "Our soldiers are blocked, the wounded are blocked. There is a humanitarian crisis ... Nevertheless, the guys are defending themselves."

Mariupol is the main port of the Donbas, a region of two provinces in the southeast which Moscow demands be fully ceded to separatists.

Ukraine says it has so far held off Russian advances elsewhere in the Donbas regions of Donetsk and Luhansk, where at least one person was killed in shelling overnight.

— Sarah Whitten