  The paper consists of two parts. In the first part we review recent work on
limit theorems for random walks in random environment (RWRE) on a strip with
jumps to the nearest layers. In the second part, we prove the quenched Local
Limit Theorem (LLT) for the position of the walk in the transient diffusive
regime. This fills an important gap in the literature. We then obtain two
corollaries of the quenched LLT. The first one is the annealed version of the
LLT on a strip. The second one is the proof of the fact that the distribution
of the environment viewed from the particle (EVFP) has a limit for a. e.
environment. In the case of the random walk with jumps to nearest neighbours in
dimension one, the latter result is a theorem of Lally \cite{L}. Since the
strip model incorporates the walks with bounded jumps on a one-dimensional
lattice, the second corollary also solves the long standing problem of
extending Lalley's result to this case.
