The Beatles formed in Liverpool in 1960 and went on to become one of the most commercially successful groups of all time. The band recorded, wrote and performed a string of number one singles and albums, and hold numerous records and accolades as one of histories most successful groups. The group's lineup consisted of John Lennon (1940-1980), Paul McCartney (1942- ), George Harrison (1943-2001) and Ringo Starr (1940- ) and all of them are still idolized and celebrated around the globe, even to this day.
There isn't a definitive answer to why the Beatles became so successful and famous, but instead a whole variety of factors contributed to their rise: The most pertinent of which were their music, image and the cultural climate of the time.
Firstly, the band's music sounded modern, catchy and radio friendly. The Beatles were one of the first true rock/pop bands, and they achieved this by writing simple, melodic pop songs that were three minutes long, and which people could sing along to. The members were all very talented musicians and wrote all of their own material (which is more than can be said for most modern pop acts). This gave them musical credibility as well as ensuring their appeal to the youth of the day. The band transformed blues and rock n roll music into the pop and rock genres which continue to be popular today; and the songs and albums themselves were extremely accessible, accomplished and catchy.
In the band's breakthrough years they were marketed as a boy band, with their target market being teenage girls - and this was reflected in the music they wrote and the clothes they wore. The Beatles' early music certainly appealed to this fan base, with the band penning hit songs such as "Love Me Do" and "I Want to Hold Your Hand", both of which cemented their boy band aesthetic and clean image. Furthermore the Beatles became fashion icons for a generation, with their suits and "mop-top" haircuts being widely copied.
The band's attitude and image struck a chord with the cultural transformation that was taking place at the time in the western world. Hippie culture had started to develop and the Beatles embraced this during the latter part of the group's existence, as was demonstrated by a change of style and musical direction. From the "Rubber Soul" album onwards, the band shed their clean-cut boy band image and instead started writing songs about drugs (instead of love), donned hippie attire and grew their hair.
As is probably the case with all successful musicians, the Beatles' phenomenal popularity probably owed something to luck and good timing, as their songs and image truly captured the zeitgeist of the 1960s - but ultimately it was the quality (and longevity) of the their music that most contributed to the the band's fame.