  Random walks provide a simple conventional model to describe various
transport processes, for example propagation of heat or diffusion of matter
through a medium. However, in many practical cases the medium is highly
irregular due to defects, impurities, fluctuations etc., and it is natural to
model this as random environment. In the random walks context, such models are
referred to as Random Walks in Random Environments (RWRE). This is a relatively
new chapter in applied probability and physics of disordered systems, initiated
in the 1970s. Early interest was motivated by some problems in biology,
crystallography and metal physics, but later applications have spread through
numerous areas. After 30 years of extensive work, RWRE remain a very active
area of research, which has already led to many surprising discoveries. The
goal of this article is to give a brief introduction to the beautiful area of
RWRE. The principal model to be discussed is a random walk with
nearest-neighbor jumps in independent identically distributed (i.i.d.) random
environment in one dimension, although we shall also comment on some extensions
and generalizations. The focus is on rigorous results; however, heuristics is
used freely to motivate the ideas and explain the approaches and proofs. In a
few cases, sketches of the proofs have been included, which should help the
reader to appreciate the flavor of results and methods.
