European Union foreign ministers have ended talks on Kosovo's declaration of independence from Serbia with member states divided on recognition.
EU foreign ministers met for what were said to be intense talks
While France confirmed that it would recognise independence, as expected, several member states led by Spain made clear their legal concerns.
US President George W Bush said Kosovo's people were "independent" but stopped short of formal recognition.
Russia has backed Serbia in its refusal to recognise Kosovo's secession.
About 10,000 students protested in Belgrade on Monday, and thousands of the city's taxi-drivers went on strike in protest at the declaration of independence, while thousands of ethnic Serbs in Kosovo's enclaves also rallied.
On Sunday, Kosovo's majority ethnic Albanians celebrated the declaration with fireworks late into the night.
Serbian security forces were driven out of Kosovo in 1999 after a Nato bombing campaign aimed at halting the violent repression of ethnic Albanian separatists.
The province has been under United Nations administration and Nato protection since then.
Sunday's declaration by the Kosovo parliament said independence would be built in accordance with the UN plan drawn up by former Finnish President Martti Ahtisaari.
Mitrovica saw a big protest by Kosovo Serbs on Monday
This outlines several limits on independence including an international presence and provision for the protection of the Serb minority.
Mr Bush, speaking in Tanzania during an Africa tour, said the plan would be the "blueprint forward".
One of the first states which recognised Kosovo was Afghanistan which referred to "the right of sovereignty".
In Brussels, the EU presidency announced after a day of intense talks between foreign ministers that member countries were free to decide individually whether to recognise Kosovo's independence.
French Foreign Minister Bernard Kouchner said after the meeting that Paris would recognise Kosovo's independence.
The issue exposed major splits within the EU, the BBC's Oana Lungescu reports.
Spanish Minister for Europe Alberto Navarro told the BBC he was frustrated that the future of Kosovo was being decided by the world's big powers in breach of international law, and said he feared it would boost separatism.
"What I say as a European is that I'm really frustrated that the future of Kosovo has been decided in Washington and to some extent in Moscow, and not in Europe," he added.
"...I think many people have many doubts about the international legality of what it is going on about this declaration of independence."
Cyprus, Greece, Romania, Bulgaria and Slovakia also expressed anxiety about the signal that recognition might send to separatists.
UK Foreign Secretary David Miliband had said it was critical for the EU to show leadership and end the cycle of violence in the Balkans.
Together with France, Germany and Italy, Britain insists Kosovo is a unique case, not a precedent, our correspondent adds.
German Chancellor Angela Merkel had said her country was seeking a "platform of unity within the EU" and would not declare its position on Monday.
The EU has already agreed to send about 2,000 police, justice and civil administration officials to oversee Kosovo and help develop the province's institutions.
Russia's parliament passed a motion on Monday condemning the declaration of independence.
Population about two million
Majority ethnic Albanian; 10% Serb
Under UN control since Nato drove out Serb forces in 1999
2,000-strong EU staff to take over from UN after independence
Nato to stay to provide security
As a permanent member of the UN Security Council, Russia can block Kosovo's entry into the organisation as a sovereign state and it said on Sunday that Kosovo's declaration should be null and void.
Serbia's interior ministry filed criminal charges on Monday against Kosovo Albanian leaders instrumental in proclaiming independence.
It accused Prime Minister Hashim Thaci and two others of proclaiming a "false state" on Serbian territory.
Chanting and playing music, students marched in Belgrade under a huge banner reading "Kosovo is a part of Serbia and Serbia is a part of the world".
The scenes were in stark contrast to riots that took place in the capital a few hours earlier when a few hundred, mainly football hooligans, went on the rampage clashing with police and stoning embassies, the BBC's Nick Hawton reports.
HAVE YOUR SAY
This is an ugly victory for demographic warfare
Some acts of violence were reported in Kosovo itself after Sunday's declaration.
A hand grenade thrown at a UN court building in the divided town of Mitrovica, and a UN car was reported to have been destroyed in the nearby village of Zubin Potok.
In the divided town of Mitrovica some chanted "This is Serbia" while others carried banners appealing for help from Russia, which has opposed Kosovo's independence.