Airports

Maastricht Aachen Airport
This is the easiest way to get to Maastricht, normally there isn't a direct flight from out the Netherlands.
Usuallly you need to go to Schiphol (Amsterdam) first.
Shuttle services available to the city (station)

Schiphol (Amsterdam)
All countries are flying to Amsterdam and it is the most easiest way to arrive.
You can choose then to go by airplane (MAA) or by train 2.5 h.
A taxi is an option also but costly (if you are by more you can share cost).

Brussels (Belgium)
All countries are flying to Brussels and it is the most easiest way to arrive.
You can choose then to go by train 1.5 h.
A taxi is an option also but costly (if you are by more you can share cost).

Other Airports.
Eindhoven-Netherlands (train or taxi) 1 h.
Luik-Belgium (taxi) 30 minutes
Dusseldorf-Germany (taxi) 1 h.
Cologne-Germany (taxi) 1 h.