Lecture 1. Random walks and Diffusions in a Random environment. General Introduction and a survey of what is known.

Lecture 2. Large deviation aspects of walks and diffusions in a random environment. The annealed and the quenched versions.

Lecture 3. Connection with Homogenization of Hamilton-Jacobi-Bellman equations with a quadratic Hamiltonian through Hopf-Cole transformation. More general equations with a convex Hamiltonian.

Lecture 4. Details of Homogenization in the time independent case. Lower bounds and the construction of super-solutions in the elliptic case for proving upper bounds.

Lecture 5. Extension to the time dependent case and the construction of parabolic super-solutions. Open problems.