During the late 16th century, when Shakespeare was getting his start, there was a boom in the theater industry. From 1576 when James Burbage and his partner built the first purpose built theater, The Theater, until the creation of The Globe, there were literally dozens of locations where acting troupes could perform their work. Early in Elizabethan times, players performed at any public house that would allow it include Inns and Play yards. In this map of 1600 London, you can see the locations of several of the theaters of London and while this map does not include all of the smaller churches and inns which might have houses plays, it shows most of the major competing playhouses that rivaled Shakespeare when he was writing for the The Globe, and the Blackfriars playhouse.
Some of the major theaters included:
The Red Lion
The Boar's Head Inn
See a full list of theaters, and where they were located (most along the River Thames) in this hand illustrated map of theaters in London 1600.
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