Putin was first elected president in 2000 and the new six-year term will keep him in power until 2018.
Putin's inauguration comes a day after an opposition protest drew more than 20,000 people, fewer than the mass demonstrations that preceded his election but still a sign that the anger over Putin's return to the Kremlin has not faded.
Police in Moscow arrested top opposition figures along with demonstrators after the protest march on the eve of Vladimir Putin's inauguration as president tried to reach the Kremlin.
The march by about 20,000 people to an island adjacent to the Kremlin proceeded peacefully this afternoon until a small group tried to break off and cross a bridge across the Moscow River, which was blocked by police.
As more people crowded toward the bridge, police sent reinforcements to the cordon, pushed demonstrators back to the rally site and began seizing demonstrators.
At the rally site, opposition leaders Sergei Udaltsov, Alexei Navalny and Boris Nemtsov were detained.
Earlier, two powerful explosions went off near a traffic police post in the capital of Russia's restive Dagestan region.
There was no definitive word on casualties.
Interior Ministry spokesman Vyacheslav Gasanov said preliminary information indicated four people were killed and three wounded.
State news agency RIA Novosti cited unnamed police sources as saying five were killed and 17 wounded while state news channel Vesti-24 said five people were taken to hospitals.
Mr Gasanov said the blasts occurred about 15 minutes apart on the outskirts of Makhachkala, but that the construction of the bombs and how they were set off was still under investigation.
Footage from the scene broadcast by state-controlled Channel 1 televison showed towering flames shooting from burning vehicles after one of the blasts.
Dagestan is plagued by frequent attacks on police by Islamist insurgents. The insurgency spread to Dagestan from neighbouring Chechnya, where Russia has fought two wars with separatists since 1994.
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