Stochastic processes in random media have become a hot topic in both mathematics and physics literature.  The random walk in random environment (RWRE) is one of the classical models to describe the behavior of some object in a solvent with impurities.  The model shows different asymptotic behavior depending on the moment condition for some parameter.  The model on one dimensional lattice is solvable, but there remain a lot of open problems for the case in higher dimensions. In this talk, we focus on the results on one dimensional lattice.  Solomon showed the law of large numbers for the transient regime.  Kesten, Kozlov and Spitzer proved the fluctuations for the transient regime and Sinai and Kesten proved for recurrent regime.  The large deviation principle was also proved for both quenched and annealed cases.  Those results will give us some clue how to treat random environment.

There are some extensions of the model.  The most popular extension is called random walk in dynamical random environment (RWDRE).  In the model we reset the environment every unit of time.  If we have time, we will consider an intermediate model between RWRE and RWDRE.