Russia's presidential election was "clearly skewed" in favour of the winner, Vladimir Putin, international monitors have said.
Mr Putin, the current prime minister, won nearly 64% of the vote. Election observers have urged any fraud allegations be fully investigated.
At least 120 people were detained at rallies in Russia, including opposition leaders Alexei Navalny and Sergei Udaltsov in Moscow.
Pro-Putin rallies have also been held.
In Moscow, police said at least 14,000 people had attended the approved protest rally on Pushkin Square, a short walk from the Kremlin. Previous protests in the city attracted up to 100,000 people.
Mr Navalny, the blogger at the heart of the election protest movement, told the rally to applause: "They will never stop. Who will stop these people? We will!"
Arguing that the protest movement in Russia's big cities had overestimated its strength and had failed to convey to the whole country the message that "thieves and swindlers" were in power, he promised a new effort.
A number of people refused to leave the square after the rally and riot police began arresting them.
Some 50 people were arrested at a separate protest on Moscow's Lubyanka Square, where the headquarters of the Federal Security Service is located.
A rally in support of Mr Putin was held on Manege Square, beside the Kremlin, with reports of about 14,000 people attending there as well.
Andrei Isayev, a senior official in the ruling United Russia party, began the rally with the slogan "Russia, Putin, victory", chanted back by the crowd.
"We have proved our right to freedom and independence," he said. "Long live the multi-ethnic people of Russia, a conqueror people."
In St Petersburg, protesters hurled flares and let off signal rockets, Russian media report. About 70 arrests were made at an unapproved rally which numbered about 800 people, police say.
Hundreds of protesters also marched in Yekaterinburg, the main city in the Urals, chanting "Putin thief!".
Amid claims of widespread fraud, the Organization for Security and Co-operation in Europe (OSCE) urged Russia to carry out a thorough investigation.
In a statement on the OSCE website,monitors said that while all candidates had been able to campaign freely, there had been "serious problems" from the start.
"The point of elections is that the outcome should be uncertain," said Tonino Picula, co-ordinator of the OSCE mission.
"This was not the case in Russia. There was no real competition and abuse of government resources ensured that the ultimate winner of the election was never in doubt."
The election was held against a backdrop of discontent, sparked by allegations of widespread fraud during the December election in favour of United Russia.
On Sunday, irregularities were seen particularly during the vote count "which was assessed negatively in almost one-third of polling stations observed", the OSCE said.
Earlier Golos, a leading Russian independent election watchdog, said it had received more than 3,000 reports of voting fraud.
The watchdog said its own exit polls suggested Mr Putin had actually won just over 50% of the vote - a result that would still have elected him in the first round.
State TV exit polls had given him about 59% of the vote.
Other allegations, including videos purporting to show evidence of voting irregularities, have been circulating online.
There were reports of "carousel voting", with voters being bussed between polling stations to cast multiple ballots, and of people being paid to vote for Mr Putin.
In other developments
- The UK acknowledged the Russian presidential election had delivered "a decisive result" in favour of Vladimir Putin despite "some problems"
- French Foreign Minister Alain Juppe said: "Despite some criticism... the re-election of President Putin is not in doubt"
- EU foreign affairs chief Catherine Ashton urged Russia to address "shortcomings" over the vote, a spokeswoman said
- Russian police detained a man allegedly planning to set off bombs at Moscow rallies on Monday and Tuesday
- Russian President Dmitry Medvedev ordered a legal review of the conviction of former tycoon Mikhail Khodorkovsky
'Glory to Russia!'
Alexei Pushkov, an adviser to Mr Putin, told the BBC that while there might have been violations, they had happened on all sides, and would not have distorted the final result.
Speaking to tens of thousands of supporters in Moscow on Sunday night, an emotional Mr Putin said he had won in an open and honest battle.
"I promised you we would win, and we won - glory to Russia!" he said.
Mr Putin moved into the post of prime minister in 2008 because of a constitutional ban on a third consecutive term as president.
He will return to the presidency in May - replacing his close ally Dmitry Medvedev - and remain in post until 2018, when he could then run for another six-year term.