Bethlem Royal Hospital

Bethlem Royal Hospital, also known as Bedlam, was a London institute for the insane.

William Shakespeare was put there when the grief he experienced after the death of his son drove him temporarily mad. With the Tenth Doctor and Martha Jones, he went there to speak with Peter Streete in 1599. (TV: The Shakespeare Code)

William Bell was confined in Bedlam asylum before, during, and after 5 April, 1761, when he expected an earthquake would destroy London and trigger the Apocalypse. (COMIC: Bêtes Noires & Dark Horses)

When asked by Ace whether Gabriel Chase was a madhouse with the patients in charge, the Seventh Doctor replied that given a chance, it could be Bedlam. (TV: Ghost Light)

Behind the scenes
According to Doctor Who: The Encyclopedia, the hospital is based in Bishopsgate.