Some 1,700 people were detained in Moscow alone, Ria news agency reports, citing the interior ministry.
The OVD-Info rights group says detentions took place in 53 cities.
Although protests have become increasingly restricted in recent years, numerous rallies have taken place across Russia since the invasion.
In the last 11 days, more than 10,000 people have been detained at protests, OVD-Info says.
"The screws are being fully tightened - essentially we are witnessing military censorship," Maria Kuznetsova, OVD-Info's spokeswoman, told Reuters news agency from Tbilisi in Georgia.
"We are seeing rather big protests today - even in Siberian cities, where we only rarely saw such numbers of arrests."
Earlier this week, government critic Alexei Navalny - who is in jail on fraud charges - called for daily demonstrations against the invasion, saying Russia should not be a "nation of frightened cowards".
However, a number of new laws have made it harder to protest in Russia in recent years, rights groups say.
"Although Russian legislation avoids explicitly using terms like 'permit' or 'ban'... it effectively requires organisers to seek authorisation for their assemblies," Amnesty International says.
According to Russian human rights group OVD-Info - which was set up in 2011 - more than 2,500 people were detained across Russia on Sunday.
It publishes the names and locations of those arrested, as well as total figures.
"Each police department may have more detainees than published lists," it says. "We publish only the names of those people about whom we know for certain and whose names we can publish."
Protests did not just take place in Russia on Sunday, but around the world. In Kazakhstan - an ally of Moscow - a peace rally was permitted in Almaty, attended by around 2,000 people.
Anti-war protesters also took to the streets in cities like Brussels, in Belgium, and London.
In Ukraine itself, Russian troops occupying the southern city of Nova-Kakhovka opened fire to try to disperse demonstrators.
A video from the city, in the Kherson region, shows protesters calling for the Russians to "go home" amid the sound of gunfire and stun grenades. The demonstrators appear to hold their ground.
One report said five people had been injured.
Photo BBC Caption: Protestors are taken away by police