The two main techniques for the generating of quasiperiodic tilings, de Bruijn’s grid method and the projection formalism, are generalized - the former to the case of any dimension and greater arbitrariness in the choice of grid and tiling vectors, the latter to the case of not necessarily cubic lattices and non-parallel grid and tiling spaces. In that way, a broad class of quasiperiodic tiling (e.g. with any desired point symmetry) can be constructed. The standard calculation of Fourier spectra is extended to the whole general class of tilings. The two generalized methods are proved to be equivalent (a constructive proof) and some applications are then discussed.