  We study branching random walks in random i.i.d. environment in $\Z^d, d \geq
1$. For this model, the population size cannot decrease, and a natural
definition of recurrence is introduced. We prove a dichotomy for
recurrence/transience, depending only on the support of the environmental law.
We give sufficient conditions for recurrence and for transience. In the
recurrent case, we study the asymptotics of the tail of the distribution of the
hitting times and prove a shape theorem for the set of lattice sites which are
visited up to a large time.
