  We compute the quasi-bound state spectra of ultralight scalar and vector
fields around rotating black holes. These spectra are determined by the
gravitational fine structure constant $\alpha$, which is the ratio of the size
of the black hole to the Compton wavelength of the field. When $\alpha$ is
small, the energy eigenvalues and instability rates can be computed
analytically. Since the solutions vary rapidly near the black hole horizon,
ordinary perturbative approximations fail and we must use matched asymptotic
expansions to determine the spectra. Our analytical treatment relies on the
separability of the equations of motion, and is therefore only applicable to
the scalar field and the electric modes of the vector field. However, for
slowly-rotating black holes, the equations for the magnetic modes can be
written in a separable form, which we exploit to derive their energy
eigenvalues and conjecture an analytic form for their instability rates. To
check our conjecture, and to extend all results to large values of $\alpha$, we
solve for the spectra numerically. We explain how to accurately and efficiently
compute these spectra, without relying on separability. This allows us to
obtain reliable results for any $\alpha \gtrsim 0.001$ and black holes of
arbitrary spin. Our results provide an essential input to the phenomenology of
boson clouds around black holes, especially when these are part of binary
systems.
