Stuttgart Airport is a fairly large international airport located 8 miles south of the city. There are direct flights to it from the UK run by British Airways and Eurowings. 

You can get from the airport to the city centry in 27 minutes on the S-Bahn (metro train) from the station directly below the terminal building. For example, by taking the S2 heading for Schorndorf.

If you're heading straight for the ground (being called the Stuttgart Arena for the Euros), you can do the journey in 53 minutes. For example, by taking the S2 as above, but then changing at Schwabstrasse onto the S1 heading for Plochingen. You then get out at Neckarpark, from where it's about a 10-minute walk to the turnstiles. More information and a journey planner can be found on the VVS Stuttgart public transport website. 

If you have a rail pass such as Interrail or the Deutschlandticket, you're covered for all the above S-Bahn journeys. If not, the VVS site has info on all other ticket options.

If you're planning to pick up a hire car from the airport, the better-known rental companies based there include Avis, Budget, Europcar and Hertz. You can see which offer the best prices here at Rentalcars.co.uk.