Around a hundred years or so after his death, serious questions began emerging regarding doubts about the true authorship of the Shakespeare plays and sonnets etc.
Since then a vast body of 'evidence' - both for and against the idea that William Shakespeare was the true author - has built up. There are hundreds of scholarly and academic articles, countless debates, dozens of books, TV documentaries etc. etc..
The question is still not resolved.
Concordia University, US, maintains a website - Shakespeare Authorship Research Center - which logs current progress.
A Declaration of Reasonable Doubt (linked to The Shakespeare Authorship Coalition) was presented to Professor William Leahy at Brunel University UK, (now at Mary Immaculate College, Ireland) by veteran Shakespearean actors Sir Derek Jacobi and Sir Mark Rylance on 8 September 2007. It now has more than 1,000 'notable' and 'academic' signatories.
Recent research projects have focused on handwriting analysis and AI-based textural analysis. Neither has provided conclusive proof one way or the other.
Substantial information on the subject, and links to scholarly references, can be found at Wikipedia