For EU citizens and travellers from countries that don't require a visa, you'll just pass straight through passport control, although you may be asked for an onward ticket (or other proof of how long you plan to spend in the country) if you're not from an EU or Schengen country.
Generally not required for stays of up to 90 days (nor for members of EU or Schengen countries). Some nationalities need a Schengen visa.
Norway is one of 26 member countries of the Schengen Convention, under which 22 EU countries (all but Bulgaria, Cyprus, Ireland, Romania and the UK) plus Iceland, Norway, Liechtenstein and Switzerland have abolished checks at common borders. The process towards integrating Bulgaria, Cyprus and Romania has slowed but they may join sometime in the future.
The visa situation for entering Norway is as follows:
Citizens of Denmark, Finland, Iceland and Sweden No visa or passport required.
Citizens or residents of other EU and Schengen countries No visa required.
Citizens or residents of Australia, Canada, Israel, Japan, New Zealand and the USA No visa required for tourist visits of up to 90 days.
Other countries Check with a Norwegian embassy or consulate.
To work or study in Norway A special visa may be required – contact a Norwegian embassy or consulate before travel.
Alcohol and tobacco are extremely expensive in Norway. To at least get you started, it's worth importing your duty-free allotment: 1L of spirits and 1L of wine (or 2L of wine), plus 2L of beer per person. Note that drinks with an alcohol content of over 60% may be treated as narcotics! You're also allowed to import 200 cigarettes duty-free. Importation of fresh food and controlled drugs is prohibited.
Svalbard is a duty-free zone; many items are considerably cheaper there than in mainland Norway as they're subject to neither MOMS (VAT) nor customs duties.
All travellers – other than citizens of Denmark, Iceland, Sweden and Finland – require a valid passport to enter Norway.