In a pre-dawn TV statement Russian President Vladimir Putin said Russia did not plan to occupy Ukraine and demanded that its military lay down their arms.
Moments later, attacks were reported on Ukrainian military targets.
Ukraine said that "Putin has launched a full-scale invasion of Ukraine".
Russian military vehicles were said to have breached the border in a number of places, in the north, south and east, including from Belarus. At least seven people are known to have been killed in Russian attacks, police say. Another 19 are missing.
President Volodymyr Zelensky announced that martial law was now being imposed across all of Ukraine.
"No panic. We're strong. We're ready for anything. We'll defeat everyone, because we are Ukraine," the Ukrainian leader said in a video statement. Ahead of Russia's attack he had made a last-ditch attempt to avert war, warning that Russia could start "a major war in Europe" and urging Russian citizens to oppose it.
Warning sirens blared across the capital, which has a population of almost three million.
Traffic queued to leave the city during the night and crowds sought shelter in the Kyiv underground. Several neighbouring countries have begun preparations to take in a large number of refugees.
"We don't understand what we should do now," one woman called Svetlana told the BBC. "We're now going to a place where we can be safe and we hope we can leave safely. We have family in Mariupol and now they're very nervous."
Russia first launched strikes on Ukraine's military infrastructure and border guard units, according to Mr Zelensky. Then Ukrainian forces said Russian military vehicles had crossed the border at Kharkiv in the north, Luhansk in the east, Russian-annexed Crimea in the south and from Belarus too.
Ukraine's army said Kyiv's Boryspil international airport was among a number of airfields that had been bombed, along with military headquarters and warehouses in the big cities of Kyiv, Dnipro, Kharkiv and Mariupol.
Mr Zelensky said Russia had positioned almost 200,000 troops and thousands of combat vehicles on Ukraine's borders.
'Unprovoked and unjustified'
The Russian leader launched the "special military operation" repeating a number of unfounded claims he has made this week, alleging that Ukraine's democratically elected government had been responsible for eight years of genocide.
He said the goal was demilitarisation and "denazification" of Ukraine. Hours earlier Ukraine's president had asked how a people who lost eight million of its citizens fighting Nazis support Nazism. "How could I be a Nazi?," said Mr Zelensky, who is himself Jewish.
Ukraine's Western allies had repeatedly warned that Russia was poised to invade, despite repeated denials from Moscow. The US, EU, UK and Japan imposed sanctions against leading Russians, Russian banks and MPs who backed the move.
Responding to Russia's assault, US President Joe Biden said Washington and its allies would respond in a united and decisive way to "an unprovoked and unjustified attack by Russian military forces" on Ukraine.
"President Putin has chosen a premeditated war that will bring a catastrophic loss of life and human suffering," Mr Biden said. "The world will hold Russia accountable."
European Commission Chief Ursula von der Leyen said the EU stood with Ukraine and that Russia "must withdraw its military". The EU's 27 leaders were due to hold an emergency summit meet later on Thursday.
UK Prime Minister Boris Johnson said he was "appalled by the horrific events in Ukraine" and that President Putin "has chosen a path of bloodshed and destruction by launching this unprovoked attack".
"Together with allies, we will respond to this barbaric act of aggression," said Czech Foreign Minister Jan Lipavsky.
Earlier this week Russia's president announced he was recognising the independence of two self-proclaimed people's republics of Donetsk and Luhansk in eastern Ukraine.
The breakaway regions were seized by Russian-backed rebels after Russia invaded Crimea in 2014. Mr Putin launched that attack after mass street protests in Ukraine that ousted pro-Russian President Viktor Yanukovych.
Since then more than 14,000 people have died in the east a conflict with Ukraine. A shaky ceasefire had held but there has been a surge in violations in recent days.
Mr Putin said the military operation's objective was to defend the people in the breakaway areas.
Kyiv and its Western allies have repeatedly rejected as absurd Mr Putin's claims that Ukraine was being run by neo-Nazis, instead pointing that - unlike an authoritarian Russia - Ukraine was now a nation with growing democratic institutions.
Fears of a Russian attack have been rising for months.
Mr Putin has repeatedly accused the US and its allies of ignoring Russia's demands to prevent Ukraine from joining Nato military alliance and offer Moscow security guarantees.
Ukraine declared a state of emergency hours before the invasion.
It introduces personal document checks, blocks military reservists from leaving the country, bans mass gatherings and places restrictions on radio communication systems. Kyiv's mayor said checkpoints would be set up in roads into the city, and access to government buildings would be restricted.
Ukraine is in Eastern Europe and shares a border with Russia. It used to be part of the Soviet Union but became an independent country in 1991