This question has been the subject of a debate for (literally) centuries. Yes, as you mentioned from the film "Anonymous", Edward de Vere was one possible candidate.
Let's start with de Vere. He was the 17th Earl of Oxford. The bulk of evidence seems to be circumstantial, though plausible. Shakespeare was not a nobleman, so he was able to write sharp-witted parody without fear of repercussion e.g. getting exiled or executed for treason against HM Queen Elizabeth I. De Vere was certainly educated and well-versed in music, law, aristocratic sports and classic as well as contemporary Italian culture, which is a big part of Shakespeare's plays. Oxford wrote poetry and prose under his own name, and it was considered very good. If he wrote the plays, they could be interpreted as political satires of Court life and critiques of government. Final facts, which are suggestive but not conclusive.
- Oxford died in 1604, and the last play was written shortly before that
This is questionable as a supporting claim for Oxford, as there is much evidence that
The Tempest was written approx 1610-1611
- In 1623, Oxford's surviving family funded the first Shakespeare folio.
In 1920, the book
Shakespeare Identified was published. It is the first modern reference to the Oxford theory of authorship.
The other candidate for authorship that I am most familiar with is Christopher Marlowe. Here is a rather focused website devoted exclusively to this subject, The Shakespeare Authorship Trust. Their list of possibilities include both de Vere and Marlowe, as well as Francis Bacon, Roger Manners, Henry Neville, Mary Sidney Herbert, William Stanley, William Shakespeare himself, or even a group theory of authorship, of one or more of that list.
The Shakespeare-Oxford Society (founded in 1957) also lists most of these individuals, as well as a timeline for why and when each was proposed as "the True Bard" and who believed it. For example, Sigmund Freud believed that de Vere was the real author, and came to that conclusion in 1926. Note that the Shakespeare-Oxford Society is
dedicated to... researching evidence that Edward de Vere (1550 – 1604) is the true author of the poems and plays of