Tens of thousands of Ukrainians have held rival pro-unity and pro-Russian rallies, at one of which, in the Crimean city of Sevastopol, pro-Russia supporters beat up their opponents.
In the eastern city of Luhansk, pro-Russian activists seized regional offices, forcing the governor to resign; and in another eastern city, Donetsk, pro-Russian protesters replaced a Ukrainian flag near the regional government building with a Russian flag.
BBC correspondent Ben Brown was at a pro-unity rally in Sevastopol which, he reports, started peacefully with demonstrators - many of them middle-aged women - waving flags and singing songs to celebrate the birth 200 years ago of Ukrainian poet Taras Shevchenko. They see him as the father of the Ukrainian language.
But by the end of the rally, pro-Russian demonstrators had turned up to gatecrash the celebrations. A line of young men and Cossacks with whips stood and glared at the rally menacingly - tension rose, and arguments broke out, both sides telling each other that Crimea is "our country".
The pro-Russians chased a group into a nearby car park and set upon the driver of a van, smashing his windscreen. He tried to drive through the mob to get away but crashed into another vehicle and was attacked again. Another person was dragged into some bushes, kicked, beaten and lashed with a Cossack's whip.
A rival pro-Russian demonstration was also staged in the city - the base of Russia's Black Sea fleet.
Not a 'single centimetre'
Addressing a huge crowd in Kiev to mark Shevchenko's birth, prime minister Arseniy Yatsenyuk pledged not to give a "single centimetre" of Ukrainian land to Russia.
However, Ukraine's defence minister has said Kiev has no plans to send the army to Crimea, where Moscow has been tightening its grip. The region is due to vote in a referendum on secession next week.
Russian president Vladimir Putin has insisted he has the right to protect Russian interests and the rights of ethnic Russians there, and says "the steps taken by Crimea's legitimate authorities are based on international law".
However, German chancellor Angela Merkel told him on Sunday that she considered the referendum illegal.
The former Russian tycoon Mikhail Khodorkovsky, who spent a decade behind bars, told a rally in Kiev's Independence Square that Moscow has been complicit with Ukraine's ousted government in using deadly violence against protesters.
Obama invites Yatsenyuk
Meanwhile, the American president , Barack Obama, has invited Ukraine's Mr Yatsenyuk to Washington for talks. The White House says the meeting this week will focus on finding a peaceful resolution to the Russian military intervention in Crimea.
Correspondents say the invitation is a significant show of support for the interim government.