Summary: The singularities that arise in polar and spherical coordinates have in the past caused significant difficulties in obtaining accurate solutions to convection-diffusion-type problems in many fields (astrophysics, geophysics, meteorology, etc.). Viewing pseudospectral methods as a limiting case of finite difference methods (rather than as based on expansions in terms of orthogonal functions) leads naturally to very simple, yet highly effective, fast Fourier transform-based pseudospectral methods in such geometries.