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Norway is part of the Schengen-cooperation. Visitors from within Schengen do not need a passport, but must be prepared to show identity card when boarding flights. A passport is always useful as an identity card in banks etc. Visitors from outside Schengen (including the UK) need passports to enter Norway (they enter the whole Schengen-area at the same time).
Even within Schengen there may be customs control. Norway only has a cooperation agreement with the European Union. Visitors are allowed to bring a small amount of alcohol tax free into Norway, at Oslo airport this can be purchased on arrival at the airport.
Although there is no imigration control between Norway and Sweden, there is still customs at the many road crossings.