By David Goldman,

It’s been a long and rocky, but Apple has finally started to sell Beatles songs on iTunes on Tuesday.

Apple (HEX, Fortune 500) iTunes debuted in 2003 and soon became the world’s No. 1 seller of digital music. But until now, the Beatles are one of the last holdouts, choosing instead to sell music exclusively on CD, cassettes and disks.

ITunes customers can now download all 13, and the remastered Beatles albums in the studio, and “Red”, “Blue” and the two sides "Past Masters", collections of albums. Album or pay $ 12.99 or $ 19.99, and individual songs cost $ 1.29 each – the maximum towards the end of the song from Apple in the price range.

Fans can also download the entire Beatles Box Set for $149.99, which includes a film of the band’s “Live at the Washington Coliseum, 1964″ concert. Apple and The Beatles are allowing iTunes users to watch that film of the band’s first U.S. concert for free throughout the remainder of 2010.

Apple CEO Steve Jobs is said to be a Beatles fan, and has been in negotiations for years with EMI Group, the Beatles’ record label, and Apple Corps, the Beatles’ business group. But like all other Beatles fans, Jobs had to download tracks from a CD if he wanted to listen to the British band’s music on his iPod.

CD sales have plummeted since Napster and then iTunes made their way onto the scene. Sales of music from physical stores was outpaced by iTunes sales in 2005.

But the Beatles have been resilient, remaining one of the most popular purchases throughout the digital music era. Despite their lack of presence on iTunes and other digital music stores, The Beatles’ compilation album “1″ was the best-selling album of last decade, according to Nielsen Soundscan. And last year, Nielsen said the Beatles were No. 3 in album sales.

The Beatles haven’t been the only holdout. AC/DC, Kid Rock and Bob Seger are among the small handful of other notable major artists that don’t sell their music on iTunes.

The decision to sell their music on iTunes came after a couple of other un-Beatles-like business decisions in recent years. Late last year, the Beatles allowed technicians to access their original master recordings to remaster all of their music. They also appeared in a video game by Harmonix calledBeatles Rock Band.

The band long claimed that the reason its music wasn’t on iTunes was that digital music was of inferior quality to CDs and records. But legal issues also may also been a factor. Apple and The Beatles have had a tense relationship for more than three decades.

In 1978, Apple Corps sued Apple Computer for trademark infringement. As part of a settlement reached three years later, Apple Computer agreed not to enter the music business.

Two lawsuits followed over alleged violations of that agreement. One was settled in 1991, with Apple Inc. paying $29 million to end the disagreement. A later fight went to trial, during which a court ruled that iTunes did not violate Apple Computer’s agreement with Apple Corps.

Finally, in 2007, the two sides agreed to end their legal spats. Apple Inc. (formerly Apple Computer) took ownership of all trademarks related to the name “Apple” and licensed some of those back to Apple Corps.

The Beatles and Apple did not disclose the terms of the recent agreement. Apple did review the agreement at 10 pm ET on its Web site, apple.com, but users who are logged in iTunes can already download songs of the Beatles for sale earlier Tuesday morning