Mr Sarkozy's family applauded his inauguration
Ceremony highlights
The new centre-right president succeeded his former mentor after defeating the Socialists' Segolene Royal in the 6 May run-off election.
In his inaugural address, Mr Sarkozy called for change and national unity.
He said his first decision was to make all schools read a letter home written by a World War II resistance fighter.
'We need results'
Mr Sarkozy said France needed "to take risks and follow initiatives".
The country also needed to "rehabilitate the values of work, effort, merit and respect" and defeat intolerance, he said.
We need to invent new solutions
Nicolas Sarkozy
Key inauguration quotes In pictures: Sarkozy's day
A 21-gun salute marked his assumption of the presidency, after a brief private conversation with Mr Chirac.
One of Mr Chirac's last tasks was to hand over the launch codes of France's nuclear arsenal.
Mr Sarkozy, 52, was accompanied at the ceremony by his wife Cecilia, wearing an eye-catching golden dress, and other family members - their son Louis and four children from their previous marriages.
In his speech Mr Sarkozy attacked racism and stressed that "order and authority is needed" to deal with security challenges.
CHIRAC'S LEGACY Unemployment 8.3%, highest for 25 years Sluggish economic growth Social tensions, especially in deprived Paris suburbs Championed failed European constitution Led French opposition to invasion of Iraq
The house that Jacques built
"We need results because the French people in their daily lives need improvements... We need to invent new solutions."
He warned that "never have the risks of inertia been so great for France" as in today's fast-moving world.
But he vowed to "defend the independence of France... the identity of France".
He also called for "a Europe that protects" and the development of "a Mediterranean union".
Mr Sarkozy later rekindled the flame on the tomb of the unknown soldier at the Arc de Triomphe.
He said he could never read the letter from a young resistance fighter, Guy Moquet - who was executed in 1941 - to his parents without being "profoundly moved".
Mr Sarkozy said it was essential children knew the horror and barbarism of war and the extent of Moquet's sacrifice.
Berlin trip
On Tuesday, Mr Chirac bade farewell to the nation in a televised address after 12 years at the Elysee.
The BBC's Paris correspondent Caroline Wyatt notes that Mr Chirac leaves behind unemployment of more than 8% and a nation uncertain of its place in the world and divided over its future.
Mr Sarkozy, formerly interior minister, has a reputation as a law and order hardliner. He made the fight against illegal immigration and crime prominent in his election campaign, along with the issue of national identity. For the first time modern France is now being ruled by an immigrant's son.
"The people conferred a mandate on me... I will scrupulously fulfil it," he promised. He won with 53% of the vote and enjoys a powerful mandate after a massive turnout by the electorate.
He praised Mr Chirac's contributions to peace and the environment. The two men have been both allies and foes over the past 30 years.
Mr Sarkozy is later to travel to Berlin to meet German Chancellor Angela Merkel, a key ally and the current president of the European Union.
Mr Sarkozy is not expected to name his prime minister and cabinet until Thursday.
A former minister, Francois Fillon, is expected be made prime minister.
Reports also suggest that a senior member of the defeated Socialist party, Bernard Kouchner, is a leading candidate for the post of foreign minister.
Élysée Palace the official residence of the President of France 
Today, the official ceremony ushering in Nicolas Sarkozy as the new president of France took place at Élysée Palace. A 21-gun salute marked the moment he assumed power from his predecessor, Jacques Chirac.

Jean-Louis Debré, president of the Constitutional Council, received the oath of office from Sarkozy. "From this day on and for the duration of your mandate, you embody France, symbolize the republic and represent all the French people," he said as he made Sarkozy the 23rd President of France.

Sarkozy's wife, Cecilia, and their five children were among the attendees.

"I will defend the independence of France. I will defend the identity of France," said Sarkozy in his inaugural address. "There is a need to unite the French people ... and to meet commitments because never before has public confidence been so shaken and so fragile."

France needs "to take risks and follow initiatives," he said, as well as "rehabilitate the values of work, effort, merit and respect."

He also said he would place defense of human rights and the fight against global warming at the center of his foreign policy.

Handing over the launch codes for France's nuclear arsenal, was one of the final tasks of Chirac's presidency. After a private meeting with Sarkozy, Chirac drove off, ending his 12 years as president, with Sarkozy waving goodbye from the courtyard of Élysée Palace. Tuesday, Chirac had made his farewells to the nation in a televised address.
Tomb of the Unknown Soldier beneath the Arc de Triomphe 
Sarkozy rode in convertible up the Avenue des Champs-Élysées to the Arc de Triomphe, where he rekindled the Tomb of the Unknown Soldier. He said that he was moved by a letter written by Guy Môquet to his parents. Môquet was executed by the Germans in 1941, along with Jean-Pierre Timbaud. It is essential that children know the horrors of war, he said. Coincidentially, Sarkozy is the first president of France to have been born after World War II.

After that, Sarkozy flew to Germany for talks about the future of the European Union.

"The first emergency is to get the European Union out of its current paralysis," Sarkozy told reporters in Berlin. "For that, it is necessary that Germany, which today holds the EU presidency, and France, which has always been its privileged partner in Europe, see eye to eye on this."