  Spin-weighted spheroidal harmonics are useful in a variety of physical
situations, including light scattering, nuclear modeling, signal processing,
electromagnetic wave propagation, black hole perturbation theory in four and
higher dimensions, quantum field theory in curved space-time and studies of
D-branes. We first review analytic and numerical calculations of their
eigenvalues and eigenfunctions in four dimensions, filling gaps in the existing
literature when necessary. Then we compute the angular dependence of the
spin-weighted spheroidal harmonics corresponding to slowly-damped quasinormal
mode frequencies of the Kerr black hole, providing numerical tables and
approximate formulas for their scalar products. Finally we present an
exhaustive analytic and numerical study of scalar spheroidal harmonics in (n+4)
dimensions.
