Ukraine's military says Russian airborne troops have landed in the eastern city of Kharkiv.
"There is an ongoing fight between the invaders and the Ukrainians," the military said in a statement on messaging app Telegram.
The battle over Kharkiv intensified on Tuesday, with Russian shelling hitting residential areas and the city's central square.
Anton Gerashchenko, adviser to the Ukrainian interior minister, said a fire broke out in the barracks of a flight school in Kharkiv following an airstrike on Wednesday.
"Practically there are no areas left in Kharkiv where an artillery shell has not yet hit," he was quoted as saying in a statement on Telegram.
The country's second largest city is home to around 1.4 million people and lies near the Russian border. It has been a target for Russian troops since last week's invasion of Ukraine began.
DW correspondent Mathias Bölinger, who is in western Ukraine, said there had been continuous attacks on Kharkiv, the capital Kyiv, and the southern city of Kherson.
"Russian troops are inside the town of Kherson," he said. "This would be the first big city in Ukraine where they seem to have taken control, although fighting is still going on, nothing is decided yet."
Bölinger also said it was not clear what the massive Russian military convoy advancing towards Kyiv would do next. "We have seen these columns standing there for some time. There are also questions about how long they can stand there because all the fuel and food that they have with them will be eaten away in the time they are standing there."
Moscow stock exchange closed for a third day
Trading on the Moscow Stock Exchange remained suspended on Wednesday, but Russia's Central Bank said it would allow a limited range of operations for the first time this week.
Western sanctions imposed over the invasion of Ukraine have sent Russia's ruble plunging to a record low. In response, the Central Bank more than doubled interest rates to 20%.
The government has also taken measures to restrict foreign investors from divesting Russian assets in a bid to stem an investor retreat.