  We study branching random walks in random environment on the $d$-dimensional
square lattice, $d \geq 1$. In this model, the environment has finite range
dependence, and the population size cannot decrease. We prove limit theorems
(laws of large numbers) for the set of lattice sites which are visited up to a
large time as well as for the local size of the population. The limiting shape
of this set is compact and convex, though the local size is given by a concave
growth exponent. Also, we obtain the law of large numbers for the logarithm of
the total number of particles in the process.
