Moscow launched dozens of air strikes across eastern Ukraine overnight, its defence ministry said on Tuesday, after Kyiv accused Russian forces of unleashing a major new offensive in the Donbas region.

Russia's defence ministry said that “high-precision air-based missiles” had hit 13 Ukrainian positions in parts of the Donbas while other air strikes “hit 60 military assets”, including in towns close to the eastern frontline.

Russia's defence minister said Moscow was seeking to “liberate” east Ukraine, but accused the West of dragging out the military operation by supplying Kyiv with arms.

“We are gradually implementing our plan to liberate the Donetsk and Lugansk People's Republics,” Sergei Shoigu said in reference to eastern Ukraine's two rebel regions, which Moscow has recognised as independent states.

“We are taking measures to restore peaceful life,” he said in a televised meeting with Russian military commanders.

Ukraine's armed forces also confirmed that fighting had increased throughout the east just hours after President Volodymyr Zelensky said Russia had kicked off the widely anticipated offensive in the Donbas region on Monday.

“The Russian occupiers intensified offensive operations along the entire line of contact,” the general staff of Ukraine's armed forces said in a report published early on Tuesday.

Ahead of the advance, Ukrainian authorities had urged people in Donbas to flee west to escape, even as officials called off evacuations for a third straight day from frontline cities due to ongoing fighting.

“No matter how many Russian soldiers are brought here, we will fight. We will defend ourselves,” Zelensky said on Telegram late on Monday.

Control of Donbas would allow Moscow to create a southern corridor to the occupied Crimean peninsula.

However, Zelensky's advisor Oleksiy Arestovich offered a slightly different take during an interview on Ukrainian TV on Tuesday, saying the offensive in the Donbas had been ongoing for nearly two weeks.

“The offensive has been going on for 12 days in the Donbas,” said Arestovich.

“They advance by looking for weak points in our defence. As soon as they find them, they infiltrate them,” he added.

Kreminna siezed

Russian forces have seized the city of Kreminna in eastern Ukraine and Ukrainian troops have withdrawn from the city, the regional governor said on Tuesday.

Kreminna, a city of more than 18,000 people about 100 km southeast of the capital Kyiv, appears to be the first city captured in a new Russian offensive in eastern Ukraine.

“Kreminna is under the control of the 'Orcs' (Russians). They have entered the city,” Serhiy Gaidai, the governor of the Luhansk region, told a briefing.

“Our defenders had to withdraw. They have entrenched themselves in new positions and continue to fight the Russian army.” He said Russian forces had attacked “from all sides”.

“It is impossible to calculate the number of dead among the civilian population. We have official statistics — about 200 dead — but in reality there are many more, ” he said, without making clear what period the estimated death toll covered.

Fresh call for Mariupol surrender

In the south, Russia continued its push to capture the besieged port city of Mariupol, as Moscow issued a fresh call for the city's defenders to surrender.

But despite the desperate situation in the city, a senior US Defence Department official on Monday said Mariupol “is still contested”.

Monday also saw the first shipments of a new $800-million US military aid package arrive at Ukraine's borders to be handed over in its fight against the Russian invasion.

'It's hell'

And while much of the focus has remained in Ukraine's east, Moscow has also targeted the country's west with air strikes, killing at least seven people in the city of Lviv near the Polish border on Monday.

Lviv has largely been spared bombardment since Russia invaded on February 24, and the city and its surroundings had become a haven for those seeking safety from the war zone.

But “today we understood clearly that we don't have any safe places in Ukraine. It's very dangerous,” a bank employee who gave her name as Natalia told AFP after the strikes.

Among the sites struck was a depot near Lviv that Moscow said held weapons recently delivered to Ukraine from the United States and Europe.

Shortly before Zelensky's address late on Monday, the regional governor of the Lugansk region Sergiy Gaiday also announced the beginning of Russia's much-anticipated attack.

“It's hell. The offensive has begun, the one we've been talking about for weeks. There's constant fighting in Rubizhne and Popasna, fighting in other peaceful cities,” he said on Facebook.

Russian shelling killed at least eight civilians in eastern Ukraine, according to local authorities.

Gaiday said four people died as they tried to flee the city of Kreminna in Lugansk as Russian troops moved in.

“The Russian army has already entered there, with a huge amount of military hardware ... Our defenders have retreated to new positions,” Gaiday said in a statement on social media.

Nearby, in the village of Novodruzhesk, weary locals have borne the brunt of the fighting.

“We are bombed everywhere. It's a miracle that we're still alive,” said Nadya, 65, her voice trembling.

“We were lying on the ground and waiting. Since February 24 we've been sleeping in the cellar.

“There's no more water, electricity, nothing. We've got no more money, no more petrol — we can't leave.”

'Heavy attacks'

Russian President Vladimir Putin has said he launched the so-called military operation on February 24 to save Russian speakers in Ukraine from a “genocide” carried out by a “neo-Nazi” regime.

Putin also recognised the independence of two self-proclaimed separatist republics in Donetsk and Lugansk shortly before the invasion began.

On Monday, Putin lauded the 64th Motor Rifle Brigade — which is accused of committing atrocities near Kyiv — bestowing battle honours on them for “heroism and valour, tenacity and courage”.

Ukraine has alleged the brigade is guilty of war crimes while occupying the suburb of Bucha on the outskirts of Kyiv, where residents were shot dead, some with their hands bound.

The European Union condemned Russia's “indiscriminate” bombing of Ukrainian civilians.

Its foreign policy chief Josep Borrell pointed to “particularly heavy attacks” in eastern and southern Ukraine and an offensive against second city Kharkiv, where officials said Russian shelling killed three people.

Seeking to strengthen ties and accelerate admission to the 27-nation bloc, Zelensky said that Ukraine hoped to receive EU candidate country status within weeks.

On Monday, he handed the EU's envoy to Kyiv a two-volume response to a membership questionnaire brought by European Commission chief Ursula von der Leyen in March.

Additional reporting from Reuters.