iTunes European stores open on song
More than 700,000 tracks are available from what Apple calls "the best music juke-box in the world".
There is no subscription fee and most albums will cost £7.99 in Britain and €9.99 in the eurozone to download.
In the US, iTunes has sold more than 85 million tracks since its launch a year ago and already dominates the market with a 70% share of all legal downloads.
From yesterday, a version was available in Britain, Germany and France. In October, Apple plans to launch an EU version of the store, available to every other European country.
iTunes has been credited with kick-starting the fightback against piracy, which is blamed for a worldwide slump in record sales.
Apple believes it can lure people away from illegal downloading by offering a superior product.
iTunes promises a speedy "one-click download", free 30-second preview of each tune and album cover art to accompany the songs.
The online store will also offer extra features such as music videos, exclusive tracks and more than 5,000 audio books from authors including John Grisham and Bill Bryson.
"It is piracy, not overt online music stores, which is our main competitor," said Apple chief executive Steve Jobs.
"You have to understand it and you have to compete with it by offering a better product.
"Illegal downloads can be unreliable, they can be slow, and they can be poor quality. Most of all, it's stealing.
"iTunes is a better product and it isn't stealing it's good karma."
The catalogue of 700,000 tracks includes 12,000 classical tunes.
In addition, there are free videos from artists including Coldplay and Jamie Cullum, live sessions from stars such as Eric Clapton and celebrity playlists from the likes of The Darkness and Moby.
While Apple claims to have signed deals with dozens of independent record labels, the Association of Independent Music said negotiations between Apple and a number of its members had broken down.
That means users will be unable to download the likes of White Stripes, Franz Ferdinand and Dizzee Rascal via iTunes.
iTunes users can listen to their music on a Mac, PC, CD or iPod. Users can burn a single song on to a CD an unlimited number of times and burn the same playlist up to seven times.
iPod sales have now passed the 3,000,000 mark and Mr Jobs says the company "can't make them fast enough" to keep up with demand.
Since the iTunes store was launched in the US last year, the music download business has grown from $135 million a year to $300m a year and is expected to be a $1.5 billion industry by 2006.
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