  Let $(Z_{n})$ be a supercritical branching process in a random environment
$\xi $, and $W$ be the limit of the normalized population size
$Z_{n}/\mathbb{E}[Z_{n}|\xi ]$. We show large and moderate deviation principles
for the sequence $\log Z_{n}$ (with appropriate normalization). For the proof,
we calculate the critical value for the existence of harmonic moments of $W$,
and show an equivalence for all the moments of $Z_{n}$. Central limit theorems
on $W-W_n$ and $\log Z_n$ are also established.
