Just to clarify my comment and on Havener's reply.
GPS is one of two, prime methods of long range navigation used for both Civil and Military flights (the other being INS, Inertial Navigation System). Aircraft do not use GPS for maintaining their vertical position, although that information is available as a cross check, they use very acurate altimeters. So much so, that we have in the last 5 years, reduced the vertical separation of aircraft above FL290 (29000ft) from 2000' to 1000', therefore at a stroke doubling our airspace capacity. (Below 300', approaching to land, Radio Altimeters are used. Very much like sonar).
However, the vast majority of the routes flown by aircraft (airways) are now marked by RNAV (GPS) fixes and not the old fashioned ground radio beacons. Therefore other than providing vertical separation, controllers can provide lateral separation knowing that if an aircraft is on the centre line of one route, it is separated from another aircraft on another route. I would estimate that over 60% of our routes and fixes are now specifically related to GPS, such that any failure of the system would bring chaos to aviation and would not be allowed to happen. To that end, Europe is well on track with its own independent GPS system.
We do still use radar for separation, particularly in the climb and descent phase, obviously always around our airports.