Mr Putin is returning to the presidency after an absence of four years in which he served as prime minister. The outgoing President, Dmitry Medvedev, was widely seen as an ally of Mr Putin.
He won a third term as president in controversial elections in March.
On Sunday, thousands of protesters opposed to the inauguration clashed with police in Moscow
Mr Putin took the presidential oath at the Grand Kremlin Palace, in a hall that was once the throne room of the Russian tsars.
In a short speech he said Russia was “entering a new phase of national development”.
“We will have to decide tasks of a new level, a new quality and scale. The coming years will be decisive for Russia’s fate for decades to come.”
He said Mr Medvedev had given a new impulse to modernisation, and the “transformation” of Russia must continue.
He also spoke of the need to strengthen Russian democracy and constitutional rights.
“I consider it to be the meaning of my whole life and my obligation to serve my fatherland and our people,” Mr Putin said.
“We will achieve our goals if we are a single, united people – if we hold our fatherland dear, strengthen Russian democracy, constitutional rights and freedoms.”
The Kremlin audience included former Soviet President Mikhail Gorbachev, former Italian Prime Minister Silvio Berlusconi, the widow of Russia’s first president, Naina Yeltsin, and the four losing presidential candidates.
If he completes his six-year term, Mr Putin will be the longest serving Russian leader since Soviet supreme ruler Joseph Stalin, says the BBC’s Steve Rosenberg in Moscow.
However, Mr Putin faces many problems; the political system he created has been showing cracks, economic growth is forecast to slow, and violence in the volatile North Caucasus continues, he adds.
The cost of Monday’s ceremony is estimated at 20m roubles (£411,000; $664,000), half of it spent on commemorative medals for the guests.
In addition, a 12m-rouble banquet will be held in honour of President Putin.
How Mr Putin deals with the wave of opposition protests which broke out last December will be a key test of his administration, correspondents say.
There was a heavy police presence for the Kremlin ceremony on Monday and some anti-Putin protesters were detained.
Sunday’s larger protest against the inauguration was peaceful until a small group of demonstrators tried to break through the lines of riot police.
Some protesters launched a sit-in by the police lines, refusing to leave unless the inauguration was cancelled.
They were also demanding an hour of TV airtime and new elections.
Prominent opposition activists Alexei Navalny, Sergei Udaltsov and Boris Nemtsov were among dozens detained.
A rival demonstration in support of Mr Putin also took place in the city.