This has been the main home of the head of the Church of England, the Archbishop of Canterbury, since 1997. If you're wondering, yes, Canterbury is a long way to commute to work each day!
As this is a private residence, it is not open to the public except by special arrangement. Archbishop's Park next to the palace is used as a public space although it is actually part of the palace grounds.
The neighbouring 14th century church, St Mary-at-Lambeth, has been converted into the Museum of Garden History.