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Ukrainian President Volodymyr Zelensky has accused Russia of war crimes after it launched air strikes on the country's second city, Kharkiv.
At least 10 people were killed and 35 hurt when an opera house, a concert hall and government offices were struck in Kharkiv's Freedom Square on Tuesday.
Later, Ukraine's military said Russian troops had parachuted into Kharkiv in an effort to capture the besieged city.
The military said there were immediate clashes after the paratroopers landed.
Earlier on Tuesday, the main TV tower in the capital Kyiv was also hit, knocking media off air and killing five people.
Moscow warned residents near Kyiv's military areas to leave their homes.
The explosion sent smoke billowing from the steel structure, but the tower remains standing.
A nearby memorial to victims of the Holocaust was damaged in the same strike. The Babyn Yar ravine is Europe's largest mass grave of the Holocaust where more than 70,000 people, mostly Jews, were shot by the Nazis.
Mr Zelensky said on Twitter that the attack was "history repeating...".
"What is the point of saying 'never again' for 80 years, if the world stays silent when a bomb drops on the same site of Babyn Yar?"
Satellite images reveal a huge Russian military convoy snaking towards Kyiv, amid fears of an all-out assault on the capital.
But a senior US defence official said there had been "no appreciable movement" by the 40-mile (64km) convoy on Tuesday.
There were indications that morale was flagging among Russian troops in general, and some units were surrendering, sometimes without a fight, the official added.
In other developments:
In a speech to the EU parliament by video link, Mr Zelensky called on Europe to "prove you are with us, prove that you will not let us go".
Later the parliament said it would look at a request by Ukraine for candidate status of the EU.
The Ukrainian president condemned the attacks on Kharkiv as "state terrorism on the part of Russia".
Video footage showed a missile hitting the local government building and exploding, causing a massive fireball and blowing out windows of surrounding buildings. Freedom Square is the second largest city-centre square in Europe and a landmark of the city.
Residents of Kharkiv told the BBC they were living in "hell" because of apparent attacks on civilians.
Yulia, a married mother of three, said the Russians were targeting areas with no military infrastructure.
"They've been bombing... a residential area where tens of thousands of people live - mothers, children," she said. "We have very many injured... This is hell."
Another resident told the Newshour programme she and her husband and children were living in a bomb shelter because they did not know where the shelling would strike next.
Joel Gunter, BBC News, Ukraine
Just 30 miles from the border and largely Russian speaking, Kharkiv has long been considered an obvious target for this invasion. Russian troops have engaged in street combat there with Ukrainian soldiers defending the city, and the city has been heavily bombarded from the air, killing scores of civilians at least.
Residents who spoke to the BBC on Tuesday described spending long nights in cramped bomb shelters or, for those with no access underground, in the hallways of their apartments, away from windows.
Iryna Ruzhynska, 40, was sheltering on Tuesday in the hallway of her second-floor apartment in a 16-floor building, with her two sons, daughter-in-law and grandchild. Ruzhynska's mother, who is 75, was trapped in her own 11th-floor apartment because the building's lift was out of order.
"We have put scotch tape on the windows and pillows by the window stills," Ruzhynska said.
Mr Zelensky's government accuses Russia of trying to lay siege to Kharkiv, Kyiv and other cities.
Russia's defence ministry said it was not targeting cities, only military infrastructure, and that there was no threat to the civilian population.
The sixth day of Russia's invasion of Ukraine saw continued attacks on several fronts, but the Russian advance has reportedly been slowed by Ukrainian resistance.
There are reports of shelling and street fighting in the southern city of Kherson, and Mariupol, a port city also in the south of Ukraine, is without electricity after an intense bombardment.
Four people, including a child, were killed when their homes were struck in Zhytomyr on Tuesday by a Russian air strike, apparently aimed at a nearby military base, a Ukrainian official said.
More than 600,000 people across Ukraine have fled their homes to escape the fighting, according to the United Nations, and Kyiv says more than 350 civilians, including 13 children, have been killed in Ukraine since the invasion began.
Ukraine has also reported that more than 70 of its soldiers have been killed, while Russia has not released figures for its own losses, amid Ukrainian claims of thousands of deaths.
"We don't turn on the lights, only the torches on our phones. We managed to go to the store yesterday, but we queued for four hours and there was virtually no food left."
A street nearby Ruzhynska's apartment was shelled on Monday, she said - a thumping blast that left body parts strewn around the street. When she opens her window she can hear gunfire and bombing.
"It is exhausting and terrifying living under this pressure," she said. "And I am angry, because this is a big, beautiful city, my city, and they want to wipe it from the face of the Earth."
