  We study the moments $M_k(T;\alpha) = \int_T^{2T} |\zeta(s,\alpha)|^{2k}\,dt$
of the Hurwitz zeta function $\zeta(s,\alpha)$ on the critical line, $s = 1/2 +
it$. We conjecture, in analogy with the Riemann zeta function, that
$M_k(T;\alpha) \sim c_k(\alpha) T (\log T)^{k^2}$. In the case of
$\alpha\in\mathbb{Q}$, we use heuristics from analytic number theory and random
matrix theory to compute $c_k(\alpha)$. In the process, we investigate moments
of products of Dirichlet $L$-functions on the critical line. We provide several
pieces of evidence for our conjectures, in particular by proving some of them
for the cases $k = 1,2$ and $\alpha \in \mathbb{Q}$.
