  A branching process in random environment $(Z_n, n \in \N)$ is a
generalization of Galton Watson processes where at each generation the
reproduction law is picked randomly. In this paper we give several results
which belong to the class of {\it large deviations}. By contrast to the
Galton-Watson case, here random environments and the branching process can
conspire to achieve atypical events such as $Z_n \le e^{cn}$ when $c$ is
smaller than the typical geometric growth rate $\bar L$ and $ Z_n \ge e^{cn}$
when $c > \bar L$. One way to obtain such an atypical rate of growth is to have
a typical realization of the branching process in an atypical sequence of
environments. This gives us a general lower bound for the rate of decrease of
their probability. When each individual leaves at least one offspring in the
next generation almost surely, we compute the exact rate function of these
events and we show that conditionally on the large deviation event, the
trajectory $t \mapsto \frac1n \log Z_{[nt]}, t\in [0,1]$ converges to a
deterministic function $f_c :[0,1] \mapsto \R_+$ in probability in the sense of
the uniform norm. The most interesting case is when $c < \bar L$ and we
authorize individuals to have only one offspring in the next generation. In
this situation, conditionally on $Z_n \le e^{cn}$, the population size stays
fixed at 1 until a time $ \sim n t_c$. After time $n t_c$ an atypical sequence
of environments let $Z_n$ grow with the appropriate rate ($\neq \bar L$) to
reach $c.$ The corresponding map $f_c(t)$ is piecewise linear and is 0 on
$[0,t_c]$ and $f_c(t) = c(t-t_c)/(1-t_c)$ on $[t_c,1].$
