Usually no planning permission is needed, but Building Regulations often apply, especially for load-bearing walls — always check first.
Listed building consent is a legal requirement for internal alterations to a listed building, and doing unauthorised work is a criminal offence.
Building work affecting the structure of a house must comply with Building Regulations, independently of any planning permission position.
The Party Wall Act 1996 requires written notice before cutting into a party wall, for example to bear a beam on it.
Councils can take enforcement action for unauthorised building work that breaches Building Regulations, and can require remedial work to be carried out.
Scotland requires a building warrant from the local authority for most structural work, and Northern Ireland has its own separate planning and building control legislation.