We study branching random walks in random environment on the
*d*-dimensional square lattice, *d*≥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, and 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.

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