Most big European cities have good flight connections to Norway and especially Oslo. Both SAS and Norwegian are continuously expanding their list of destinations both in and outside Europe.
The Irish airline Ryanair (www.ryanair.com) also offers cheap flights from several European cities to Oslo Rygge (about an hour outside Oslo) and Haugesund (1,5 hour drive from Stavanger and 3 hours from Bergen) . Easyjet also recently started flying to Bergen from London Gatwick.
Norwegian recently started an Oslo – New York flight and the prices are not at all bad! On the ‘right’ days and dates you can get tickets down to as little as NOK 1000 one way.
They will also start flying to other destinations in the US in the near future and are planning to keep the prices nice and low. SAS is also worth checking out for reasonable flights from the US and Asia to Oslo.