  Let $(Z_n)$ be a supercritical branching process in a random environment $%
\zeta$, and $W$ be the limit of the normalized population size
$Z_n/\mathbb{E%}(Z_n|\zeta)$. We show necessary and sufficient conditions for
the existence of weighted moments of $W$ of the form $\E W^{\alpha}\ell(W)$,
where $\alpha\geq 1$, $\ell$ is a positive function slowly varying at $\infty$.
In the Galton-Watson case, the results improve those of Bingham and Doney
(1974).
