BBC Television Centre (or "TVC" to meeja types) was opened in June 1960, and has been home to a great deal of BBC Output since then.
Back in 2007 it seemed like Television Centre's days were numbered, when it was scheduled to be sold off by the Beeb by 2013. However, more recently the credit crunch has meant it simply isn't be worth as much, so it looks as though the BBC will retain it.
Interestingly, plans to sell off the building have led to it potentially being scheduled for listed status by English Heritage, who had previously agreed not to list it unless the BBC disposed of it, so the BBC could make adjustments, so perhaps their indecision may cause them problems in the future.
The BBC offer an award-winning tour of Television Centre six days a week, Monday to Saturday. Booking is essential, however, and it costs £9.30. Clearly the telly tax doesn't stretch this far. You can find out more about the tours at http://www.bbc.co.uk/tours/details/details_tvc.shtml and they can be booked on 0370 901 1227.