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Note: flight, boat, train and bus timetables change constantly, and airlines come and go, so please do not rely solely on this information for your travel planning. Check with relevant companies, or a flight search engine like Skyscanner, first.
There are 4 airports within striking distance of Stockholm:
Arlanda Airport (40km north) is the largest, handling the majority of international flights. Its fast rail link into the city (20 minutes to Central Station) makes it very handy, too.
Bromma Airport (8km west) is the closest to central Stockholm, but actually no faster in terms of transfer time (coaches reach City Terminal in 20 minutes; local buses take 30 minutes). It handles mostly domestic traffic plus a few international flights.
Skavsta Airport (100km south) is used chiefly by low-cost airlines and is about an hour by car or train from Stockholm, or 90 minutes by the Flygbussarna bus. You could tie in a visit to the town of Nyköping (pronounced new-shopping), just 7km away.
Västerås Airport (110km west) has become popular with low-cost carriers as an alternative route to the capital, or for those combining Stockholm with nature-tourism or the historic city of Västerås. The Flygbussarna bus takes about 90 minutes to reach Stockholm's Central Station.
Flights from the UK:
Services include British Airways from London Heathrow and London City to Arlanda; SAS from London Heathrow, Bristol (summer only) and Edinburgh to Arlanda; Norwegian Air Shuttle from London Gatwick and Manchester to Arlanda; and Ryanair from London Stansted to Skavsta and Västerås.
Flights from the rest of Europe:
Major airlines serving Arlanda include Aer Lingus, Air Berlin, Air France, Iberia, KLM, Lufthansa, Vueling and TAP Portugal. There are also flights from numerous European cities to Arlanda with SAS and Norwegian Air Shuttle, and to Skavsta with Ryanair and Wizz Air.
From further afield:
Arlanda is served by Delta flights from New York JFK, Emirates flights from Dubai, Thai Airways flights from Bangkok, Qatar Airways flights from Doha, SAS flights from Chicago and Newark, and Norwegian Air Shuttle flights from Los Angeles, New York JFK and Fort Lauderdale. Otherwise, your best bet is to fly via another European hub such as London Heathrow or Copenhagen.
BY PUBLIC TRANSPORT
Stockholm's metro system, the T-bana, is quick and convenient. There are 3 lines - red, blue and green - which between them cover most of the main areas (useful stops include Gamla Stan, plus Slussen on Södermalm). All lines intersect at Central Station.
The city also has a good bus network, and a tram which runs from Norrmalm to Djurgården.
There's a comprehensive network of ferries in and around the city. Most routes in central Stockholm operate year-round; those serving the archipelago are generally confined to the summer months. Perhaps the most useful for tourists is the Djurgårdsfärjan service between Slussen, Skeppsholmen and Djurgården. See Stromma Skargard and Waxholmsbolaget for details of other routes.
Otherwise, Stockholm’s main islands are easy to stroll between. For the best views, head north from Slussen along the eastern side of Gamla Stan, then along Strandvägen to leafy Djurgården.