  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
focus on the supercritical case, when the process survives with a positive
probability and grows exponentially fast on the nonextinction set. Our main is
goal is establish Fourier techniques for this model, which allow to obtain a
number of precise estimates related to limit theorems. As a consequence we
provide new results concerning central limit theorem, Edgeworth expansions and
renewal theorem for $\log Z_n$.
