The earliest known protest occurred outside Russia's embassy in Washington about 1am on Thursday (1700 AEDT), only three hours after President Vladimir Putin said he had launched his military operation.
Local news reports showed dozens of protesters in the US capital waving Ukrainian flags and chanting "Stop Russian aggression!"
In London, hundreds of demonstrators, many of them Ukrainian and some weeping, gathered outside Downing Street, home to the prime minister, urging Britain to do more.
"We need help, we need someone to support us," said one. "Ukraine is too small and the pressure is too big."
In Paris, one demonstrator told Reuters: "I feel that we are in a very dangerous moment for the whole world."
In Madrid, Oscar-winning Spanish actor Javier Bardem, nominated for another Academy Award this year, joined about a hundred protesters outside the Russian embassy.
"It is an invasion. ... It violates Ukraine's fundamental right to territorial sovereignty, international law, and many other things," Bardem said.
A giant flag was carried through Manhattan's Times Square by a crowd of several hundred protesters.
In the Swiss capital Bern, hundreds gathered, holding Ukrainian flags and chanting "Peace for Ukraine!".
A small demonstration in Geneva, organised by the Nobel Peace Prize-winning International Campaign to Abolish Nuclear Weapons (ICAN) outside the U.N. European headquarters, condemned what the group said was Putin's threat to use nuclear weapons.
Other demonstrations were held in Beirut, Tel Aviv, Dublin and Prague.
Also in Dublin, a Russian double-eagle crest beside the gate of the Russian embassy was defaced with red paint.
More protests were scheduled for later in the day in the U.S. cities of Houston and Denver, according to social media posts.
In Russia itself, protesters defied an official warning that explicitly threatened criminal prosecution and even jail time for those calling for or taking part in protests.
Hundreds rallied in 53 cities including Moscow, St Petersburg and Yekaterinburg, chanting slogans such as "No to war!" and holding up makeshift signs.
By Thursday evening, police had detained no fewer than 1667 people in 53 cities, the OVD-Info rights monitor said. Six hundred were arrested in Moscow alone, the Tass news agency reported.
Protesters defied a warning issued on Thursday by the Investigative Committee, a kind of Russian answer to the FBI, that explicitly threatened criminal action and even jail time for people calling for or taking part in protests.
Russia's telecommunications regulator warned media organisations on Thursday not to circulate what it described as "false information" about Moscow's massive military operation against Ukraine and threatened to block offending content.
