EUFOR in Bosnia consists of about 7,000 troops from 33 countries with some not being EU members, such as Canada and Turkey.
The command of EUFOR was taken over by British General David Leakey, while the last commander of the NATO-led Stabilisation Force (SFOR), Steven Schook, became the first commander of the NATO headquarters in the Bosnian capital consisting of some 150 officers from NATO member states.
Present at the Sarajevo ceremony were NATO Secretary-General Jaap de Hoop Scheffer, the EU High Representative for the Common Foreign and Security Policy, Javier Solana, NATO's chief commander in Europe, US General James Jones, and his deputy John Reith, who will also have command authority over EUFOR.
The current chairman of Bosnia's three-man presidency, Borislav Paravac, said that this was a significant moment not only for Bosnia but also for the entire international community.
"I welcome the arrival of the EU peace mission and offer it full support and cooperation. I also welcome the continuation of NATO presence which will help in the reform of the national defence system, attempts to arrest war crimes suspects, as well as in the fight against terrorism," Paravac said.
Scheffer confirmed that efforts would be made to continue providing full support to the International Criminal Tribunal for the former Yugoslavia (ICTY) in cooperation with the EU and with the assistance of NATO's headquarters in Sarajevo.
Whether Bosnia will become a member of NATO's Partnership for Peace programme depends on cooperation between local authorities in Bosnia and the ICTY, the NATO Secretary-General said.
The EU is assuming a new responsibility with the same determination and efficiency shown by NATO, Javier Solana said.
He went on to say that the common goal was to see Bosnia within the European bloc one day.
This will depend on the readiness of local political leaders to strengthen Bosnian state institutions and the country's economy and democracy, he said adding that the mission would continue until its accomplishment.
After the ceremony Scheffer and Solana held a news conference at which they voiced confidence that the new military mission in Bosnia would be as successful as the previous one led by NATO.
Solana reiterated that Bosnia's destiny was to be a member of the EU.
Answering questions about efforts to arrest war crimes fugitives, wanted by the Hague tribunal, Scheffer and Solana said that it remained to be the primary responsibility of local authorities in Bosnia, but EUFOR and NATO's headquarters in Sarajevo were willing to fully support the search for indictees.
Scheffer said that it was up to people in Bosnia to decide how long NATO's headquarters in Bosnia would remain.
The duration of the EUFOR mission is a matter of a broader cooperation between the Bosnian authorities and the EU, Solana added.