Having endured long battles to capture cities in the neighbouring Lugansk region, Russia is now seeking to push deeper into Donetsk to consolidate its hold over the entire Donbas region.
Air raid sirens sounded overnight throughout the country's east and south.
Residents in the small town of Druzhkivka, south of the eastern Ukrainian industrial city of Kramatorsk, woke up to a suspected missile attack on Saturday which ripped apart a supermarket and left a massive crater outside.
Ukrainian officials said on Saturday five people were killed in the Donetsk region in the past 24 hours while seven were injured.
"The entire frontline is under relentless shelling," Donetsk military administration chief, Pavlo Kyrylenko said in a Telegram message on Friday night.
He said the city of Sloviansk, on which Moscow's troops have now set their sights, is being "shelled day and night".
He also accused Russian forces of setting agricultural fields on fire, saying they were "trying to destroy the harvest by all means".
Lugansk governor Sergiy Gaiday, said on Saturday the Russians were attacking Donetsk from their bases in his region.
"We are trying to contain their armed formations along the entire frontline... Where it is inconvenient for them to go forward, they create real hell, shelling the territories on the horizon," he said.
Kyrylenko warned the Russians were in the process of replenishing troops in the region to prepare for further assaults.
Oleksandr Vilkul, the mayor of President Volodymyr Zelensky's hometown of Kryvyi Rih in central Ukraine, said on Saturday Russia had attacked the city with cluster munitions, killing at least one person and injuring two.
Zelensky said in his nightly address on Friday he had spent the day on the frontlines in the eastern Dnipropetrovsk region, where he met civilian and military leaders.
"The eyes of all aggressive political movements and regimes in the world are now focused on what Russia is doing against us, against Ukraine," Zelensky said Saturday in an Instagram post.
"Will the world be able to bring real war criminals to justice?" he asked, warning failure to do so would lead to "hundreds of other aggressions".
But in a Telegram message on Saturday, an official from the region's military administration warned Russia had "intentionally shelled residential areas", and had not stopped "terrorising" cities and villages.
The Ukrainian general staff said the majority of bombardments took place in east Ukraine and the second-largest city of Kharkiv but there was no ground offensive.
In the south, the mayor of Mykolaiv begged citizens not to leave shelters, as he said explosions were heard throughout the night.
Ukraine's Deputy Prime Minister Iryna Vereshchuk was quoted by Ukrainian media as urging people in occupied areas to evacuate by any means possible.
"Massive fighting is going to happen," she said.
Kharkiv governor Oleg Sinegubov on Saturday said four people were injured in attacks on the region, adding that the Russians were "engaged in defensive actions".
'Further Evolution Of Support'
In a boost to Kyiv, Washington announced $400 million of further military aid, including a type of artillery ammunition with "greater precision", and that has previously not been sent.
"It's a further evolution in our support for Ukraine in this battle in the Donbas," a senior defence official was quoted by the US Department of Defense as saying.
Also included in the aid package are four additional High Mobility Artillery Rocket Systems to add to eight already in place.
"From a security assistance perspective, this is a steady drumbeat now, and it is a long-term commitment to Ukraine," the same official was quoted as saying.
Britain's defence ministry on Saturday said the first group of up to 10,000 inexperienced Ukrainian military recruits began training in England as part of a UK-led programme.
US Urges China To Condemn 'Aggression'
The United States also put pressure on Russia diplomatically at a meeting of Group of 20 foreign ministers in Indonesia.
Washington and allies condemned Russia's assault ahead of the gathering before Russian Foreign Minister Sergei Lavrov faced what US Secretary of State Antony Blinken called a barrage of Western criticism at the closed-door talks.
Blinken on Saturday called for China to distance itself from Russia after talks with his Chinese counterpart in Indonesia.
Blinken said he told Wang Yi "this really is a moment where we all have to stand up, as we heard country after country in the G20 do, to condemn the aggression, to demand among other things that Russia allow access to food that is stuck in Ukraine".
He added there were "no signs" Moscow was willing to engage after the G20 talks.
"If there is an opportunity for diplomacy, we will seize it," he said.
Lavrov was defiant Friday and accused Western nations of avoiding "talking about global economic issues" instead of the war.