When did Shakespeare's theatre start?
It depends what you mean by "Shakespeare's theatre". Do you mean the theatre which was built in 1996 and is called Shakespeare's Globe Theatre? Or do you mean the theatre company which he joined, the Lord Chamberlain's Men, which was founded in 1594? Or do you mean the tradition of Elizabethan theatre of which Shakespeare was a part, which started in about 1560? It could be any one.
They were most commonly presented at the Globe Theatre but only between 1599 and 1642. Later theatres which have seen Shakespeare's plays even more often include Covent Garden, Drury Lane, the Old Vic, the National Theatre, the Royal Shakespeare Theatre, and the Festival Theatre in Stratford Ontario.