A general formulation is presented for the ENDOR spectra of a randomly oriented, polycrystalline (powder) S = 1 2 paramagnet, assuming slow cross-relaxation, g anisotropy domination of the EPR spectrum, arbitrary symmetries and relative orientations of g and hyperfine tensors, and δ-function EPR and ENDOR component lineshapes. Calculations for simple, archetypical types of centers have been presented, as well as selected calculations showing the full generality of the method. In particular, at g values where the ENDOR spectra arise from multiple orientations of the paramagnet (powder-type spectra) it was found that the ENDOR shape functions, N(A)g, show two divergences and no steps when the g and A tensors are coaxial; noncoaxiality can introduce subsidiary maxima and can cause the occurrence of as many as four additional divergences. © 1984.