Transportation from Stockholm-Bromma Airport to your destination: 

Stockholm Public Transit:
SL Bus 110 runs to Alvik, where the metro train (Tunnelbana) or the streetcar stations are located;  and
SL Bus 152  runs to Alvsjo Station (Stockholm's International Fair) via Solna & Sundbybeg.

Airport coaches travel directly between the Airport and the City Terminal in about 20 minutes or so.

A new light railway (Tvairbanan) to the City
is 1 km from the terminal
(but outside of airport boundaries).  It connects to the Central railway station within 20 minutes.

Transfer to either Arlanda or Skavsta Airport by Flygbussarna:
  -  to Stockholm-Arlanda Airport: 
Flygbussarna now serves Terminals 2, 4 & 5 with departures every 20 minutes, with main stops at Kista, Sundbyberg & Bromma - thereby creating a cross-town connection in Stockholm, with more passengers & commuters being able to take mass transit to /from the airports.  Buses are equipped with WiFi (free) & electrical outlets.
  -  to Stockholm-Skavsta Airport transfer. Flybussarna also serves Skavsta Airport.
For more information visit:  flygbussarna.se

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