In a broadcast to the nation, Volodymyr Zelenskyy said: “We are defending our country, we fight for our country and we protect our country.”
Mr Zelenskyy, 44, said weapons would be issued to any Ukrainian who wanted one and appealed for blood donations for wounded soldiers.
The enemy had suffered “serious losses”, he added.
Ukraine has cut diplomatic ties with Russia, Mr Zelenskyy said, and introduced martial law in all territories.
Advising people to stay at home, he said “we are strong, we are ready for everything”.
Russian President Vladimir Putin said he had ordered a “specialised military operation”, heralding the beginning of the invasion.
Sky News correspondents heard explosions in Kyiv and Kharkiv and understands that borders have been breached in at least four areas of Ukraine.
Ukraine’s foreign minister, Dmytro Kuleba, tweeted that Mr Putin had “just launched a full-scale invasion of Ukraine”.