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The importance of the area of Durham containing its castle and cathedral has been recognized by UNESCO as a World Heritage Site: see Link
To quote UNESCO:
"Durham Cathedral was built in the late 11th and early 12th centuries to house the relics of St Cuthbert (evangelizer of Northumbria) and the Venerable Bede. It attests to the importance of the early Benedictine monastic community and is the largest and finest example of Norman architecture in England. The innovative audacity of its vaulting foreshadowed Gothic architecture. Behind the cathedral stands the castle, an ancient Norman fortress which was the residence of the prince-bishops of Durham."

See also
EH World Heritage Site listing: Link (includes useful map)
EH Grade I listings:
The Cathedral: Link
Cloister (West): Link
Cloister (South): Link
Cloister (East): Link
Lavatorium: Link
Deanery: Link
Prior's Kitchen: Link
Priory Prison: Link
12-15 The College: Link
Chorister School: Link
8 The College: Link
Wall to west of 13, South Bailey: Link
Castle Gatehouse: Link
Castle Keep: Link
Castle North Range: Link
Castle West Range: Link
Castle Wall (W,N & E): Link
Castle Terrace Wall: Link

UNESCO World Site List: Link

See other images of Durham Castle & Cathedral UNESCO World Heritage Site