Kosovo's parliament has unanimously endorsed a declaration of independence from Serbia, in a historic session.
Celebrations went on into the night after Prime Minister Hashim Thaci promised a democracy that respected the rights of all ethnic communities.
Serbia's PM denounced the US for helping create a "false state".
A split later emerged at the Security Council, when Russia said there was no basis for changing a 1999 resolution which handed Kosovo to the UN.
Seven Western countries said it was quite clear the situation had moved on.
Tens of thousands of people had thronged the streets of Kosovo's capital, Pristina, since the morning.
When news came of the declaration in parliament, the centre of the city erupted with fireworks, firecrackers and celebratory gunfire.
Crowds surrounded an independence monument which was unveiled during the evening and signed by Mr Thaci and Kosovo President Fatmir Sejdiu.
Ethnic Albanians staged noisy celebrations in the Macedonian capital, Skopje, and in Brussels, outside the headquarters of Nato and the European Union.
The first sign of trouble in Kosovo came in the ethnic Serbian area of the flashpoint town of Mitrovica, where two hand grenades were thrown at international community buildings.
One exploded at a UN court building while the other failed to go off outside offices expected to house the new EU mission.
In Belgrade, demonstrators threw stones and broke windows at the US embassy as riot police tried to fend off a crowd of around 1,000 people.
The protesters, described as gangs of youths, also attacked a McDonald's restaurant, the Serbian government building and the embassy of Slovenia which currently holds the EU presidency.
Several Serbian ministers had travelled to Kosovo to show their support for the ethnic Serbian minority.
Kosovo's 10 Serbian MPs boycotted the assembly session in protest at the declaration.
Serbia's Prime Minister Vojislav Kostunica blamed the US which he said was "ready to violate the international order for its own military interests".
"Today, this policy of force thinks that it has triumphed by establishing a false state," Mr Kostunica said.
Search for equality
The declaration was approved with a show of hands. No-one opposed it.
"We have waited for this day for a very long time," Mr Thaci told parliament before reading the text, paying tribute to those who had died on the road to independence.
From today, he said, Kosovo was "proud, independent and free".
"The independence of Kosovo marks the end of the dissolution of the former Yugoslavia," the prime minister said.
He said Kosovo would be built in accordance with the UN plan drawn up by former Finnish President, Martti Ahtisaari.
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Unfortunately today Kosovo and Serbia are to become two dispensable chess-pieces of EU/NATO and Russia
Mat, Ljubljana, Slovenia
The international military and civilian presence - also envisaged by the Ahtisaari plan - was welcome, the PM said.
There should be no fear of discrimination in new Kosovo, he said, vowing to eradicate any such practices.
The declaration was signed by all the MPs present.
The UN Security Council went into emergency session on Sunday evening after Russia called for the United Nations to declare the Kosovo declaration illegal.
UN Secretary General Ban Ki-moon called on all sides to keep to their commitments and refrain from violence.
Russia's UN ambassador Vitaly Churkin told reporters that the resolution allowing the UN to administer Kosovo since 1999 was still in force so there could be no legal basis for any change in status.
But seven Western states said the UN Security Council could not agree on Kosovo's future and all attempt to reach a negotiated outcome had been exhausted.
"We regret that the Security Council cannot agree on the way forward, but this impasse has been clear for many months," Belgium's UN ambassador Johan Verbeke said.
He gave the statement on behalf of Belgium, France, Italy, the UK, Croatia, Germany, and the United States.
Limitations of independence
The declaration approved by Kosovo's parliament contains limitations on Kosovan independence as outlined in Mr Ahtisaari's plan.
Kosovo, or part of it, cannot join any other country. It will be supervised by an international presence. Its armed forces will be limited and it will make strong provisions for Serb minority protection.
Recognition by a number of EU states, including the UK and other major countries, will come on Monday after a meeting of EU foreign ministers in Brussels, says the BBC's Paul Reynolds.
The US is also expected to announce its recognition on Monday.
Three EU states - Cyprus, Romania and Slovakia - have told other EU governments that they will not recognise Kosovo, says our correspondent.
Russia's foreign ministry has indicated that Western recognition of an independent Kosovo could have implications for the Georgian breakaway provinces of Abkhazia and South Ossetia.