By Air

SAS have a scheduled service from most major Airports including Dublin, Manchester, Zurich New York, Chicago and London(LHR) to Arlanda(ARN) Stockholm International Airport. They also fly connecting fights through Copenhagen(CPH). Fares start at €175, £99 plus taxes from Ireland and the UK.
Connections to Stockholm are by Express train (20 mins to the Central Station), Flyingbus, or Taxi.

Ryanair fly to 2 smaller regional airports about 75km from Stockholm: Skavsta(NYO) from Stansted(STN) and Glasgow Prestwick(PIK) and Vesterås(VST) from Stansted(STN).
Connections to Stockholm are by Flyingbus, Train or Car. Both airports are just over an hour from Stockholm.

BMI, British Airways, Swiss, Lufthansa, KLM also fly to Arlanda.