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There are different options - Hamburg airport (HAM) is north of the city centre but relatively close.
A taxi will be between 20-30 Euro depending on where you want to go. These stop right outside the arrivals hall and are always in plentiful supply.
Second option is to buy an HVV (Hamburg public transport system - see separate section) ticket - single journey is 3 Euro (2014). The train station is located directly outside the airport terminal buildings.
The journey to or from the airport takes 25 minutes (one-way) and trains depart every 10 minutes. If you want to transfer from train to the underground line, you should change at Ohlsdorf, which is a junction station for U-Bahn (underground) and S-Bahn (train) lines.
It is the cheapest option, especially if you want to do further travelling on the HVV network as you can buy a day/multi-day ticket.