The protest was peaceful until a small group of demonstrators tried to break through the lines of riot police.
Opposition activists Alexei Navalny, Sergei Udaltsov and Boris Nemtsov have all been detained.
A rival demonstration in support of Vladimir Putin has also been taking place.
Organizers said about 20,000 people took part in the opposition march – to an island close to the Kremlin – although police put the figure at about 8,000.
Alexei Navalny urged protesters not to disperse until those arrested had been released.
Speaking to a radio station by phone from a police van, Alexei Navalny also told the protesters to insist that the authorities carry out the reforms they have promised.
Police have been blocking the protesters from crossing a bridge over the Moscow River.
Clashes broke out when more people crowded towards the bridge and riot police wielding batons pushed demonstrators back towards the rally site, witnesses said.
Protesters launched a sit-in by the police lines.
They were refusing to leave unless Vladimir Putin’s inauguration was cancelled. They were also demanding an hour of TV airtime and new elections, our correspondent says.
Dozens of protesters are said to have been arrested.
TV images then showed police storming the stage of the rally where left-wing activist Sergei Udaltsov had been addressing protesters, and taking him to a waiting vehicle.
Before he was seized, Sergei Udaltsov had called for the inauguration to be cancelled and said through a loud-hailer: “We will not leave.”
The crowd chanted back: “We are the power.”
Russia’s Interfax news agency later reported that Alexei Navalny – an anti-corruption blogger and nationalist – had been arrested and Dozhd TV tweeted that liberal politician Boris Nemtsov had also been detained at the rally.
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