The charismatic, Western-backed Mikhail Saakashvili has begun his second term as the president of the former Soviet republic of Georgia.
Mr Saakashvili's next challenge is parliamentary elections
After being sworn in he met Russia's foreign minister, with both countries saying they wanted improved ties.
Thousands of opposition protesters held a rally claiming the 6 January poll was rigged, even though international observers said it was democratic.
Mr Saakashvili saw his support almost cut in half at the elections.
His radical free-market reforms during his first term of office have caused widespread discontent.
Thousands took to the streets in January after Mr Saakashvili polled 53% of the vote in the presidential race, narrowly averting a run-off against his nearest rival, Levan Gachechiladze, who won 25%.
Mr Saakashvili called the snap poll to resolve a crisis after suppressing anti-government rallies in November last year.
Following the inauguration, Mr Saakashvili's opponents held their largest demonstration yet against his re-election.
Reuters reports as many as 80,000 people took part in the protest, although this figure could not be confirmed.
The BBC's Matthew Collin, in Georgia, says Mr Saakashvili is no longer the seemingly unchallengeable figure who was swept to power by the Rose Revolution in Georgia, four years ago.
The opposition wants the election results to be overturned
Following his inauguration ceremony on Sunday he told the crowd:
"We held the most democratic elections in Georgia's history and in this election you made the choice for the unity of Georgia and the democratic development of Georgia."
Mr Saakashvili has admitted that his attempt to salvage Georgia's devastated economy led to a difficult period of change.
Mr Saakashvili has responded by promising to cut unemployment, increase pensions and introduce new social welfare programmes.
But he also wants to push forward with his bid to join Nato.
Georgia's chances were damaged when he used force to break up opposition protests two months ago.
Russia remains strongly opposed to Georgia's bid to join the Western military alliance.
This could lead to yet more disputes between Mr Saakashvili's government and the Kremlin, although both sides have said they want to improve ties.
During his inauguration speech Mr Saakashvili said that Georgia's desire to integrate with the West did not run counter to the interests of its neighbours.
"I again extend a hand of partnership to our northern neighbour [Russia]. We should be friends, we should be closer and we should stand together," he said.
Russian Foreign Minister Sergei Lavrov attended Mr Saakashvili's inauguration and the two men met briefly afterwards.
Mr Lavrov said Russia's decision to send such a high-ranking official was of "no small importance".
"Our participation in the Georgian president's inauguration ceremony confirms Russia's sincere and deep intention to normalize relations with Georgia," Interfax news agency quotes him as saying.
But Mr Saakashvili's immediate challenge is at home. The Georgian opposition insists his presidency is not legitimate.
Our correspondent says parliamentary elections are due in the spring, which means the scene is set for what could be a turbulent start to Mr Saakashvili's second term in office.