Luxembourg airport (LUX) is served by many national airlines and is 6km east of the city. The national airline Luxair covers many European countries: http://www.luxair.lu/
Cityjet flies to/from a number of UK cities, including London: http://www.cityjet.com/
Easyjet flies from/to London and Milan since last year: http://www.easyjet.com/en
Another alternative is Ryanair, flying to Frankfurt-Hahn airport (HHN) from many European cities. This airport is around 120km from Luxembourg and is connected by a regular bus service (http://www.flibco.com/schedules/metz_lux_hahn), which takes 1h40m.
Luxembourg is reached from Brussels and the North West by the E411; from Paris and the West by the A4 (E50); from Metz and the South via the E25, and from Trier and Koblenz to the East by the E44.
Direct trains from Paris take just over 2 hours and Brussels around 3 hours. There are also direct trains from Bonn, Cologne, Dusseldorf and several other European cities.