Hollande, 57, is the seventh president of the Fifth Republic and the first Socialist to hold the office since Francois Mitterrand.
"We are a single France, undivided," he said after his investiture ceremony. "My mandate is to bring France back to justice, open up a new way in Europe, contribute to world peace and preserve the planet."
Hollande promised a presidency of "dignity, simplicity and soberness."
Hollande met with outgoing leader Nicolas Sarkozy and received the country's nuclear codes. The two men were then joined by Hollande's partner Valerie Trierweiler and Sarkozy's wife, Carla Bruni-Sarkozy.
The new president has a series of stops to make in Paris Tuesday, including a visit to City Hall, where Jean-Marc Ayrault is expected to be named prime minister, The New York Times said.
Hollande will then fly to Berlin to dine with German Chancellor Angela Merkel. The Times said the relationship between the heads of the two biggest economies in the eurozone will be vital to the future of the European Union.