As perhaps might be obvious from the use of quotation marks above, I believe the author of William Shakespeare's plays and poems was in fact...William Shakespeare. Obvious? Hardly worthy of the word "controversy"? Rather startlingly, among some this remains a point of real contention.
Being something of a collector of conspiracy theories--and feeling qualified to opine at length about this one--herein you'll find my twin pennies on the subject...
For many years, a variety of individuals (some of them quite prominent persons, with considerable accomplishment) have doubted that Shakespeare wrote Shakespeare. Some simply look at the scarcity of hard data we have about the matter and express doubt. All well and good. None can deny the lives of Elizabethan playwrights are mostly matters of mystery, with many records that no doubt existed having gone the way of all things in the ensuing centuries. We're not even sure about his birthdate, birth certificates being pretty much unheard-of at the time. Others, however, take a further tack and insist that from what we do know of Shakespeare he could not have been the author. More, they usually have specific candidates in mind.
Much of the argument against Shakespeare, let us be frank, remains extremely subjective. How, some dissidents argue, could the son of an illiterate glove-maker from a provincial town like Stratford-Upon-Avon possibly written the greatest poetry in English literature? A man who didn't even attend University! They point to the plays and how various people in (for example) politics or the law or the church insist those words show an "insider's" knowledge and feel. But at its heart, this argument rests on snobbery. I for one would find zero difficulty believing a great writer might not be noble, or that they might never have gone to a major university. For the one, I reject there is something inherently superior in those lucky enough to have had successful ancestors. And for the other, does the premise not suggest that no great literary artist could have arisen prior to the invention of the university? Well, what about Euripedes, Sophocles, Aeschylus, Aristophanes and the like? More, I would posit that genius has its own power. Stephen Crane many times in his life was approached by those insisting he must have actually seen combat in the Civil War because no one who had not could possibly have written The Red Badge of Courage. Of course, Crane was born five years after that war ended! Likewise there've been accusations that a "nobody" like J.K.Rowling could not have written such a vastly popular work as Harry Potter, that instead a secret committee must have done it and the publishers hired Ms.Rowling to masquerade as the books' author. I've not heard anyone make a similar claim about Stephanie Meyer but that is probably just a matter of time.
Mind you, there's a bigger problem that this matter of opinion regarding the Author's knowledge. Quite simply, the Author of the plays (whoever it might have been) showed a less-than-perfect knowledge of certain subjects, such as an understanding of what was genuinely believed to be the nature of the heavens. More subtly, careful scholarship reveals that the Author's sources for much of his material (regarding the history plays, for example) are few in number. Indeed, the impression of a broad but not extremely deep knowledge is created--at least that was my impression (which is as valid as any Anti-Stratfordian, surely).
They are on somewhat stronger ground when pointing to the sheer vocabulary shown in the plays, gigantic by any measure and exceeding that of the King James Bible! But again, surely the Author was a genius and is that not something one might expect of genius? Still stronger evidence is that of William Shakespeare's last will and testament, or at least at first glance. That document--which does survive--makes no mention of books or shares in the acting company of which Shakespeare was part owner. By any measure this seems odd. But then, consider what else is omitted. Famously, Shakespeare left his "second best bed" to his widow. Nowhere is a "best bed" mentioned. Neither is any kind of manuscript or document mentioned. Is it likely the man had no papers at all? Or--and this is offered as a theory, nothing more--are we applying our own expectations to a document rather than the desires and circumstances of the man who actually wrote it?
In fact, there are quite a lot of mysteries surrounding Shakespeare's life, whether he wrote those plays or not. But lack of knowledge is not the same as proof. In fact, the two are very nearly opposites. And the mysteries of Shakeaspeare's life are few and far between compared to those surrounding virtually any other playwright of his country and age.
Now, look at who the Anti-Stratfordians (i.e. those who insist Shakespeare did not write Shakespeare) posit as being the true Author. The current favorite is Edward de Vere, the Earl of Oxford. They insist he's noble and University educated, with the wit and poetic ability to have composed those plays. But their theory falls apart on several key points. First--why keep it a secret? Oxford was a rebel, the English Renaissance version of a hippie or rogue. Why not proclaim his authorship and get the glory (since it is acknowledged that the plays were very popular)? Second--he died in 1604 and evidence indicates that the plays continued to be produced for years afterwards. What "Oxfordians" usually claim is that such evidence is inconclusive. But they don't apply the same criteria to their own ideas.
More tellingly, at least IMHO, is the lengths to which supporters of Christopher Marlowe as the Author go. He died in 1593, the victim of a fight in a tavern with a dagger through his eye. Rather than claim the plays were not produced following 1593 (which would be ridiculous), these adherents claim Marlowe only faked his death as part of some intelligence scheme hatched by Queen Elizabeth's spymaster Sir Francis Walsingham (who, however, died three years previously). Of course, unlike Oxford, Marlowe's works are already acknowledged as works of great literary merit. But they are also agreed to be of a radically different style than that of Shakespeare's.
To give a concrete example--Marlowe's works show an increased use of rhyming couplets, wherein subsequent lines do rhyme with one another ("For never was a story of more woe/Than this of Juliet and her Romeo" from Romeo and Juliet). Yet Shakespeare showed the exact opposite pattern, reducing the number of rhyming couplets in favor of "hidden" rhyme and variations in meter.
And then we have some other facts that Anti-Stratfordians really have trouble defeating--such as contemporary comments about Shakespeare and his plays, including published references to Shakespeare as the author. One or two or even three may be dismissed, but there are far more than three. In the end, doesn't Occam's Razor indicate that the simplest explanation is probably the mainstream one--that a genius of a slightly distinctive family (his father was the equivalent of mayor at one point), with the typical education of a boy of his class of the time, went on to have a successful career in Elizabethan theatre, writing these wonderful works? No need for elaborate conspiracies about faked deaths or secret identities or reworking chronologies. Just genius, which we already know is part of the truth of this story.