Russia is trying to take full control of the Donbas, comprised of two eastern provinces Moscow claims on behalf of separatists.
Moscow has poured thousands of troops into the region, attacking from three sides in an attempt to encircle Ukrainian forces holding out in the city of Sievierodonetsk and its twin Lysychansk. Their fall would leave the whole of Luhansk region under Russian control, a key Kremlin war aim.
"All the remaining strength of the Russian army is now concentrated on this region," President Volodymyr Zelenskiy said in a late night address.
His office said the Russians had launched their assault on Sievierodonetsk early on Wednesday and the town was under constant mortar fire.
Luhansk regional governor Serhiy Gaidai said six civilians had been killed and at least eight wounded, most near bomb shelters, in Sievierodonetsk. The main road out was still being shelled, he said, but humanitarian aid was still getting in.
Ukraine's military said fighting for the road was ongoing, and that on Tuesday it had repelled nine Russian attacks in the Donbas. It reported at least 14 civilians killed in strikes by aircraft, artillery, tanks, mortars and missiles.
In Pokrovsk, a Ukrainian-held Donbas city that has become a major hub for supplies and evacuations, a missile had blasted a crater in a railway track and damaged nearby buildings.
In Kramatorsk, nearer the front line, streets were largely deserted, while in Sloviansk further west, many residents took advantage of what Ukraine said was a break in the Russian assault to leave.
"My house was bombed, I have nothing," said Vera Safronova, seated in a train carriage among the evacuees.
Russia is also targeting southern Ukraine, where officials said shelling had killed a civilian and damaged scores of houses in Zaporozhzhia and missiles had destroyed an industrial facility in Kryviy Rih.
In the latest sign of Moscow's plans to solidify its grip on territory it has seized, President Vladimir Putin signed a decree simplifying the process for residents of newly captured districts to acquire Russian citizenship and passports.
In a speech by video link to dignitaries at a global forum in Davos, Switzerland, Zelenskiy said the conflict could only be ended with direct talks between him and Putin.
As a "first step towards talks", Russia should withdraw to lines in place before its Feb. 24 invasion, he said. Prior to the invasion, Russia held Ukraine's Crimea peninsula, while its separatist proxies occupied parts of the Donbas.
Ukraine's closest allies say they fear some Western nations might push Kyiv to give up land for peace. Estonia's prime minister said Ukraine should not be forced into compromises.
"It is much more dangerous giving in to Putin than provoking him. All these seemingly small concessions to the aggressor lead to big wars. We have done this mistake already three times: Georgia, Crimea and Donbas."