  We study the asymptotic behaviour of random walks in i.i.d. random
environments on $\Z^d$. The environments need not be elliptic, so some steps
may not be available to the random walker. We prove a monotonicity result for
the velocity (when it exists) for any 2-valued environment, and show that this
does not hold for 3-valued environments without additional assumptions. We give
a proof of directional transience and the existence of positive speeds under
strong, but non-trivial conditions on the distribution of the environment. Our
results include generalisations (to the non-elliptic setting) of 0-1 laws for
directional transience, and in 2-dimensions the existence of a deterministic
limiting velocity.
