Whilst it has never been proven that the Beatles did have any illegitimate children, there have always been rumors that this may be the case. After all, they were four young good-looking guys in a vastly popular band touring the country, and were known to have many female "admirers".
As far as the rumors go, Paul McCartney certainly seems to be the Beatle who has attracted the most speculation. It was well-known that McCartney was a hit with the ladies and John Lennon once described Paul as a "sex gladiator". Various stories concerning alleged illegitimate children of Paul's have surfaced for years but none seems more genuine or believable than the case of Anita Cochrane.
Anita Cochrane had been a friend and fan of the band since their early days playing at the Cavern Club in Liverpool. It has been well-documented (and never denied by McCartney) that they spent the night together at least three times. In 1963 Cochrane had fallen pregnant (although she also admits spending the night with another man at around the same time) and she gave birth to her son Phillip on the 10th of February 1964.
Following this, and after numerous attempts by Anita to contact Paul, Brian Epstein, the band's manager settled out of court with Anita for a sum of £5000 pounds. A clause in the agreement that she must renounce all claims that Paul was the father was also included in the contract. From then on, the Cochrane case had been successfully kept quiet until a British newspaper broke the story in 1997 - at which point Anita's son, Phillip (who was now 33) demanded a paternity test to find out the truth. However, the three parties were allowed to work out their problems privately, and no further statements have been made to the press since then. It is still not clear whether a DNA test was ever carried out.