Since the verified facts of Shakespeare’s life are so few, even a person normally skeptical of conspiracy theories could well consider it possible that the plays and poetry bearing the name of William Shakespeare were written by A. N. Other. But just who could have been that other?
Well, even with few verified facts about Shakespeare’s life, we can know some facts about the author of these texts by reading the texts themselves.  Whoever was the author must have spent a lot of time hanging about with actors, since knowledge of, and in-jokes about, acting and the theatre permeate the plays.  Also, whoever it was must have grown up in a rural district, not in a big city, since the author of the plays and the poetry knows a great deal about animals and plants, about rural life and its myths and customs, and rural pursuits.  Whoever it was also had close connections to Warwickshire, since the plays contain words specific to that area.
Also, whoever it was must have had close personal or family connections to the old religion (Catholicism), since many of the plays make detailed reference to, or indeed seem to be allegories of, the religious differences of the time (Wilson 2004, Asquith 2005). Whoever it was was close enough to the English court to write plays which discussed current political issues using historically-relevant allegories, yet not so close that these plays themselves or their performances (with just one exception) were seen as interventions in court intrigues.
Whoever it was also knew well the samizdat poetry of Robert Southwell, poet and Jesuit martyr, since some of the poetry and plays respond directly to Southwell’s poetry and prose (Wilson 2004, Klause 2008). To have responded to Southwell’s writing before 1595, as the writer of Shakespeare’s narrative poems and early plays did, required access to Southwell’s unpublished, illegal, dissident manuscripts.  Southwell and Shakespeare were cousins (Klause 2008 has a family tree).
And finally whoever it was was not a playwright or poet already known to us, since these texts differ stylistically from all other written work of the period, while exhibiting strong stylistic similarity among themselves.
There is only one candidate who fits all these criteria, and his name is William Shakespeare. Anyone seriously proposing an alternative to Shakespeare as the author of Shakespeare’s plays and poetry needs to explain how that person could have written poetry and plays with all the features described above. Every alternative theory so far advanced – Kit Marlowe, the Earl of Oxford, Francis Bacon, Elizabeth I, et al. – falls at the factual hurdles created by the texts themselves.
Note: Klause [2008, p. 40] presents a genealogy which shows that Robert Southwell and William Shakespeare shared a great-great-great-great-great-grandfather, Sir Robert Belknap (c. 1330-1401, Chief Justice of the Court of Common Pleas of England, 1377-1388) – Southwell through his mother, Bridget Copley, and Shakespeare through his mother, Mary Arden.  In addition, the great-great-grandfather, Sir John Gage, of Shakespeare’s patron, Henry Wriothesley, Third Earl of Southampton, was also grandfather to Edward Gage, husband of Margaret Shelley, Southwell’s mother’s first cousin and, like his mother, a descendant of Sir Robert Belknap.  In the extended families of Elizabethan society, all three – Shakespeare, Southwell and Wriothesley – would have been seen as, and would have known each other as, cousins.   The bonds across such extended family relationships were strong.   Having lived in contemporary societies (in Southern Africa) where extended families still play a prominent role (Bourdillon 1976), the strong loyalty and close brotherhood engendered across such apparently-distant connections is perfectly understandable to me, if not yet to all Shakespeare scholars.
References:
Clare Asquith [2005]: Shadowplay: The Hidden Beliefs and Coded Politics of William Shakespeare.  UK: Public Affairs.
Michael F. Bourdillon [1976]: The Shona Peoples: An Ethnography of the Contemporary Shona, with Special Reference to their Religion. Shona Heritage Series. Gwelo, Rhodesia (now Gweru, Zimbabwe):  Mambo Press.
John Klause [2008]:  Shakespeare, the Earl and the Jesuit.  Madison, NJ, USA: Fairleigh Dickinson University Press.
Anne R. Sweeney [2006]: Robert Southwell: Snow in Arcadia:  Redrawing the English Lyric Landscape 1586-1595. Manchester, UK:  Manchester University Press.
Richard Wilson [2004]: Secret Shakespeare:  Studies in Theatre, Religion and Resistance. Manchester, UK:  Manchester University Press.