  We consider the branching process in random environment $\{Z_n\}_{n\geq 0}$,
which is a~population growth process where individuals reproduce independently
of each other with the reproduction law randomly picked at each generation. We
describe precise asymptotics of upper large deviations, i.e. $\mathbb{P}[Z_n >
e^{\rho n}]$. Moreover in the subcritical case, under the Cram\'er condition on
the mean of the reproduction law, we investigate large deviations-type
estimates for the first passage time of the branching process in question and
its total population size.
