Shakespeare Globe Theatre and the national theatre (NT) are the two most popular repertory theatre companys in London. ('repertory theatre' is when a selection of shows are rotated in the same venue every few days so you can see several shows in one week)
The National Theatre (NT) is a subsidised organisation, which basically gives it license to take risks with new and sometimes unknown writers, and productions that may sometimes not appeal to the wider audience, however the most successful NT productions do tend to transfer into the West End if it is deemed commercially viable (War Horse being a prime example!)
The Royal National theatre complex features three theatres, the Olivier, the Lyttelton and the Dorfman, and with the majority of performances being repertory it is possible see several plays in the same week. The season for the Shakespeare Globe Theatre runs between April and September, (due to most of the venue being open air) and of course generally performs Shakespearean plays.