We analyse the angular eigenfunctions-spin-weighted spheroidal harmonics-and eigenvalues of Teukolsky's equation. This equation describes infinitesimal scalar, electromagnetic and gravitational perturbations of rotating (Kerr) black holes. We derive analytic expressions for the eigenvalues up to sixth order in the expansion parameter for low frequencies and an analogous expansion in the high-frequency limit. Spin-weighted spheroidal harmonics form a complete and orthonormal set of functions on a prolate spheroid. They are, however, not the eigenfunctions of the natural Laplace operator on a spheroid and thus do not allow an obvious geometrical interpretation as the corresponding spin-weighted spherical harmonics.