Ukraine's President Volodymyr Zelensky has said Russia has launched an assault to seize the eastern Donbas region.
Moscow bombarded cities with rocket and artillery fire on Monday and in a video address Mr Zelensky said that the battle "for the Donbas has begun".
Ukraine's top security official, Oleksiy Danilov, said that Russia tried to break through the Ukrainian front lines in the region, reports BBC.
The offensive has been long-expected after Russia failed to seize Kyiv.
Russia initially appeared to want to capture major Ukrainian cities and topple the government.
But after facing stiff resistance, Russian defence officials said that its main objectives in the "first stage of the operation" had been "generally accomplished" and its forces were moved from areas around the capital.
They announced plans to redirect the focus of the invasion towards the "liberation" of the Russian speaking Donbas region.
Russia's President Vladimir Putin has portrayed the invasion as an attempt to demilitarise and "denazify" Ukraine, something Ukraine and its allies dismiss as a ruse for an unprovoked attack.
Throughout Monday, Russia unleashed a barrage of rocket and artillery fire on a number of eastern areas, with eight civilians killed in the city of Kreminna in Luhansk and in the Donetsk area.
Seven people were killed and 11 more were injured in four Russian strikes in Lviv, a western city that has largely been spared the attacks seen elsewhere in Ukraine.
The governor of the Luhansk region said the situation was "hell", with constant fighting being reported in some cities.
In Ukraine's second city of Kharkiv, the regional governor said evacuations of civilians were taking place in areas where intense clashes are expected.
Russian defence officials said it its forces had hit hundreds of military targets in Ukraine on Sunday night, including 16 military facilities in the Kharkiv, Zaporizhzhia, Donetsk and Dnipropetrovsk regions, as well as a port in Mykolayiv in the south and east of Ukraine.
Speaking in a video message on Monday night, Mr Zelensky said that he and his forces "will defend ourselves" and pledged "not give up anything Ukrainian".
"A very large part of the entire Russian army is now focused on this offensive," he added, but said that Ukraine's forces will fight on "no matter how many Russian troops are driven there, we will fight".
Moscow claimed last month that it controlled 93 per cent of Luhansk and 54 per cent of Donetsk and its forces are expected to try and encircle the remaining Ukrainian troops in the region.
But they face a protracted fight with some of Kyiv's most battle-hardened troops. Ukraine is believed to have between 40-50,000 soldiers in the Donbas, many of whom have spent years fighting against Russian-backed separatists forces in the region.
Zelensky's chief of staff, Andriy Yermak, said the assault marked "the second phase of the war" and assured Ukrainians that their forces could hold off the offensive.
"Believe in our army, it is very strong," he said.
The change in Russia's objectives as set out by President Vladimir Putin during a speech last week, where he announced that his aim was "to help the people who live in the Donbas, who feel their unbreakable bond with Russia".
The Pentagon's press secretary, John Kirby, cautioned Russia could be "setting the conditions for future offensive operations" and officials in Kyiv warned that the new Russian offensive does not mean that Moscow has ended its attacks on other parts of Ukraine.
"Putin hasn't removed the goal to destroy us as a state and our political leadership," Mr Danilov said in an interview with Ukrainian TV.
Russia continues to target the south-eastern city of Mariupol, which would allow its troops to complete a land bridge between the occupied Crimean peninsula and forces in separatist-held regions of eastern Ukraine.
Officials in Kyiv claimed Russian war planes were preparing to drop five-tonne bombs on the Azovstal plant where the final Ukrainian holdouts are sheltering.
