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The American, British, French, Saudi and Kuwaiti aircraft took off at 2330 GMT last night.
Their bombs were aimed at military and strategic targets, including an oil refinery and Baghdad airport.
At least 400 raids took place. Latest reports say all the Allied aircraft have returned home safely, although France says four of its planes were hit.
One Tornado was shot down
US Defence Secretary, Dick Cheney, said the operation appeared to have gone "very well".
Two hours after the raids began, President George Bush made a televised address.
He said the military objectives were clear - force Iraqi troops out of Kuwait and restore the legitimate government.
In Baghdad, Saddam Hussein remained defiant. He said the "Mother of all Battles had begun". He urged the Iraqi people to "stand up to evil".
First news of the bombing came from reporters in Baghdad working for the American TV network, CNN. They reported hearing air raid sirens shortly before the bombs hit.
President Bush said: "Our operations are designed to best protect the lives of all the coalition forces by targeting Saddam's vast military arsenal.
"Initial reports from General Schwarzkopf are that our operations are proceeding according to plan."
The British Prime Minister, John Major, came out of Number Ten shortly before 0800 GMT to make a statement to reporters.
"No-one wanted this conflict. No-one can be pleased about the fact this conflict has been necessary," he said.
"I hope now it is clear to Iraq that the scale of the Allied operation is such that they cannot win.
"I hope that Saddam Hussein will now make a very swift decision that he will do what he's been invited to do by the world community for a long time, that he should get out of Kuwait and end this matter swiftly and decisively."
He said the attacks would continue until Saddam withdrew his troops.
Allied planes have taken off this morning to launch a second round of air strikes.
Iraq had refused to comply with a UN ultimatum for its troops to withdraw from Kuwait by 15 January, so at 2330 GMT on 16 January 1991, Operation Desert Storm began.
The Allies' devastating and sustained aerial bombardment involved cruise missiles launched from US warships. US, British and Saudi Arabian fighter planes, bombers and helicopters were also involved.
After more than a month of intensive air attacks, the Allies launched a land offensive on 24 February. One day later, the Iraqis began retreating.
On 27 February, President George Bush declared victory. Kuwait was liberated but Saddam Hussein remained in power in Baghdad.
On 20 March 2003, President George Bush junior led a "coalition of the willing" on an invasion of Iraq with the aim of toppling the Bagdad regime.
Within a month, Saddam Hussein had been ousted and the Americans were claiming victory in the war.
Saddam Hussein was captured after several months in hiding in December 2003. He was tried by an Iraqi court, sentenced to death and executed on 30 December 2006.
US-led coalition forces ran the country until power was officially handed back on 28 June 2004.
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