When investigating quantum or semiclassical phenomena in a general relativistic context one usually resorts to quantum field theory in curved spacetime. However, traditional quantization methods rely on foliating spacetime into spacelike hypersurfaces or on fixing asymptotic boundary conditions in time. This is a serious limitation if for the problem of interest no suitable Cauchy hypersurfaces exist and/or if boundary conditions are naturally given on timelike hypersurfaces, either at finite locations or asymptotically. I will outline methods of quantization adapted to such situations and the conceptual insights on which they are based. If time permits I will comment on how this may contribute to the search for a quantum theory of gravity.