From the longer Wikipedia page 
The Shakespeare authorship question is the argument about whether someone other than William Shakespeare of Stratford-upon-Avon wrote the works attributed to him. Anti-Stratfordians—a collective term for adherents of the various alternative-authorship theories—say that Shakespeare of Stratford was a front to shield the identity of the real author or authors, who for some reason did not want or could not accept public credit. Although the idea has attracted much public interest, all but a few Shakespeare scholars and literary historians consider it a fringe belief and for the most part disregard it except to rebut or disparage the claims.
Shakespeare's authorship was first questioned in the middle of the 19th century, when adulation of Shakespeare as the greatest writer of all time had become widespread. Shakespeare's biography, particularly his humble origins and obscure life, seemed incompatible with his poetic eminence and his reputation for genius, arousing suspicion that Shakespeare might not have written the works attributed to him. The controversy has since spawned a vast body of literature, and 80 authorship candidates have been proposed, including Francis Bacon, the 6th Earl of Derby, Christopher Marlowe, and the 17th Earl of Oxford.
Comment on the disputeEdit
It should be noted that the concept of 'authorship', particularly of plays, at the time was much more fluid than it is now - the plays were working documents, and Shakespeare made use of a number of existing stories for his plays, and collaborated with other authors.