Arundel is a market town and civil parish in the Arun District of the South Downs, West Sussex, England. The much-conserved town has a me…Arundel is a market town and civil parish in the Arun District of the South Downs, West Sussex, England. The much-conserved town has a medieval castle and Roman Catholic cathedral. Arundel has a museum and comes second behind much larger Chichester in its number of listed buildings in West Sussex. The River Arun runs through the eastern side of the town.