  For any odd prime $\ell$, let $h_\ell(-d)$ denote the $\ell$-part of the
class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-d})$.
Nontrivial pointwise upper bounds are known only for $\ell =3$; nontrivial
upper bounds for averages of $h_\ell(-d)$ have previously been known only for
$\ell =3,5$. In this paper we prove nontrivial upper bounds for the average of
$h_\ell(-d)$ for all primes $\ell \geq 7$, as well as nontrivial upper bounds
for certain higher moments for all primes $\ell \geq 3$.
