Live updates: follow the latest news on Russia-Ukraine
New shelling and attacks have sent soaring numbers of refugees fleeing, sometimes under fire, as the death toll continues to grow.
Russia's attacks on Ukraine, now in their 11th day, have caused more than 1.5 million people to flee the country in what the UN has called "Europe's fastest growing refugee crisis since World War Two".
Pope Francis on Sunday deplored the "rivers of blood and tears" flowing in Ukraine, as Washington said there were "very credible reports" that Russia had committed war crimes by deliberately attacking civilians.
Hundreds of civilians have been killed and thousands wounded, with hundreds of thousands of people, mainly women and children, pouring into neighbouring countries such as Poland, Romania or Moldova.
Efforts on Saturday to move people out of Mariupol, the scene of some of the war's greatest ferocity, collapsed almost immediately, with both sides accusing each other of breaching a ceasefire agreement.
The next attempt, on Sunday, also failed with the warring parties again exchanging recriminations.
But the Governor of the eastern region Donetsk, Pavlo Kirilenko, said "the column to evacuate the population could not leave Mariupol" because Russian forces "started to bombard the city".
Very few refugees from the strategic city on the Azov Sea made it out on Saturday, but one family arrived in the central city of Dnipro and recounted their harrowing experience.
"We stayed in the basement for seven days with no heating, electricity or internet and ran out of food and water," one family member said.
"On the road, we saw there were bodies everywhere, Russians and Ukrainians … we saw that people had been buried in their basements."
Disturbing scenes from the fighting have filled social media.
The New York Times posted a particularly gruesome photo on its website showing what it said were a mother and two children killed by Russian shelling outside Kyiv.
Ukraine's Prime Minister Denys Shmyhal urged the Group of Seven countries to expel Russia and Belarus from the International Monetary Fund and all World Bank organisations to further isolate Mr Putin.
Western allies have imposed unprecedented sanctions against businesses, banks and billionaires in a bid to choke the Russian economy and pressure Moscow to halt its assault.
Further punitive action, including a possible ban on Russian oil imports, could be imposed if Mr Putin does not change course, world leaders warned.
US officials said on Sunday that they were in "active discussions" with European nations about such a ban.
But Mr Putin has equated global sanctions with a declaration of war and warned that Kyiv is "putting in question the future of Ukrainian statehood".
Russia would reach its aims of "neutralisation" of Ukraine "either through negotiation or through war", Mr Putin told Mr Macron on Sunday, an Elysee official said.
Nato allies have so far rebuffed Ukraine's calls for a no-fly zone, with one senior US senator, Marco Rubio, saying on Sunday that it could lead to a "World War Three" against nuclear-armed Russia.
Mr Putin has threatened "colossal and catastrophic consequences, not only for Europe but also the whole world" if a no-fly zone were set up.
In the latest sign that sanctions were biting, Moscow said on Sunday that retailers in Russia would restrict sales of essential goods including bread, rice and flour to limit black-market speculation.
Payment giant American Express on Sunday stopped operations in Russia, a day after Visa and Mastercard announced similar steps.
As more people took to the streets across Russia against the military assault on Ukraine, Moscow intensified a crackdown on dissent, detaining more than 2,500 protesters.
The Ukrainian military said on Sunday that it was engaged in "fierce battles" with Russian forces near the southern city of Mykolaiv and Chernihiv in the north.
"The main efforts are focused on defending the city of Mariupol," it said in a Facebook post, adding an operation by Ukrainian forces was also under way in the eastern part of the Donetsk region.
In the eastern city of Kharkiv, a television tower has been hit in an air strike, interrupting broadcasts, regional Governor Oleg Synegubov said.
Elsewhere, Russian forces have been inching closer to the capital Kyiv in an assault that has become ever-more indiscriminate and deadly.
At Bilogordoka on the outskirts of Kyiv, Ukrainian troops have planted explosives on what they say is the last intact bridge standing in the way of advancing Russian forces.
"This is the last bridge," said a fighter who gave his name as Casper. "We're defending it and not letting them break through to Kyiv."
The former paratrooper, who joined Ukraine's volunteer territorial defence, said his team would blow up the bridge if the Russians advanced, and "sink as many enemy tanks as we can while we do it".
Dozens of civilians have been killed in the northern city of Chernihiv. Some survivors have been reduced to living in craters or amid mounds of debris.
"There were corpses all over the ground," a man who gave his name only as Sergei told AFP, as air raid sirens wailed.
"They were queueing here for the pharmacy that's just there, and they're all dead."
Moscow has insisted that it is not aiming at civilian areas.
Rejecting its denials, US Secretary of State Antony Blinken told CNN that "we've seen very credible reports of deliberate attacks on civilians, which would constitute a war crime".
Ukraine's President, Volodymyr Zelenskyy, warned the Russians were turning their attention to the historic western port city of Odesa, which has so far been largely spared.
Kyiv has urged the West to boost its military assistance, with Mr Zelenskyy pleading for Eastern European neighbours to provide Russian-made planes that his pilots are trained to fly.
A barrage of Russian missiles destroyed an airport in central Ukraine's Vinnytsia, Mr Zelenskyy said.
Mr Blinken said Washington was "working actively" on a deal with Poland to supply it with American jets.
Washington reportedly is working on a deal for Poland to send Soviet-era aircraft to Ukraine in return for US fighter jets.
Weapons, ammunition and funds have poured into Ukraine from western allies as they seek to strengthen Kyiv. Washington last week authorised $350 million of military equipment.
Tens of thousands of people demonstrated over the weekend in cities from London to Barcelona to Washington in support of Ukraine.
Twenty thousand international volunteers have travelled to Ukraine to join in the fight, Foreign Minister Dmytro Kuleba said, although a British official cautioned that doing so would be "unlawful" for Britons.
The invasion has reverberated through the cultural world as well.
The music director and principal conductor at the Bolshoi Theatre in Moscow, Tugan Sokhiev, became the latest high-profile figure to quit his post over his loyalties.
A third round of Russian-Ukrainian talks aimed at finding a way out of the bloody conflict is set for Monday, Ukraine said on Saturday.
Diplomatic talks have continued, with Mr Zelenskyy saying on Sunday he spoke by telephone with US President Joe Biden, Italian Prime Minister Mario Draghi and British Prime Minister Boris Johnson, to discuss further support for his country and sanctions against Russia.
Turkish President Recep Tayyip Erdogan spoke with Mr Putin on Sunday, urging him to agree to "an urgent general ceasefire".