  Let $\mathbb B_n$ be the open unit ball in $\mathbb C^n$. We characterize the
spectra of pointwise multipliers $M_u$ acting on Banach spaces of analytic
functions on $\mathbb B_n$ satisfying some general conditions. These spaces
include Bergman-Sobolev spaces $A^p_{\alpha,\beta}$, Bloch-type spaces
$\mathcal B_\alpha$, weighted Hardy spaces $H^p_w$ with Muckenhoupt weights and
Hardy-Sobolev Hilbert spaces $H^2_\beta$. Moreover, we describe the essential
spectra of multipliers in most of the aforementioned spaces, in particular, in
those spaces for which the set of multipliers is a subset of the ball algebra.
