His name was on the title pages of a number of plays published in his lifetime. He was described as an excellent playwright by dozens of people alive at the same time he was, who moved in the same social circles and must have known him. The connections between the plays, the Chamberlain's/King's Men, the Globe Theatre Sharers and William Shakespeare, gent. of Stratford are demonstrated by a mountain of documentary evidence. For example, Shakespeare's will leaves a bequest to the same two men (Heminges and Condell) who published the First Folio, and are named as actors in the same company as Shakespeare in payment receipts for the King's Men, who are shown on the covers of the published plays to be the only company to perform some of them, and were also the same people who owned the Globe Theatre where some people kept diaries saying that they attended performances of these same plays. All of this evidence consistently points to the same people being associated in all these endeavors and to the fact that the person credited with writing the plays is William Shakespeare, gent., of Stratford. There is no documentary evidence whatsoever that anyone other than Shakespeare wrote those plays, or that any of Shakespeare's contemporaries doubted for one moment that Shakespeare was the author.
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William Shakespeare
William Shakespeare wrote the Tempest in 1611.
Since you have added this question to the William Shakespeare catedgory, you won't be surprised to find that these three plays were all written by William Shakespeare, the most famous playwright ever.
William Shakespeare
The Stratfordian argument is that William Shakespeare wrote the majority of the plays and sonnets in his repertoire. The Anti-Stratfordian argument is that he did not write what he was associated with but an aristocrat did instead.