Random walks are among the most fundamental models for stochastic evolutions, with applications ranging from the description of physical motions in homogeneous or disordered media to the modeling of macromolecules or explorations of networks in data science. The principal objectives of this course are to introduce techniques to study random walks on general weighted (potentially random) graphs, and to investigate the profound connections between the behavior of random walks and the underlying graph topology using tools from potential theory, geometric group theory, and electrical networks. Later parts of this course cover selected advanced topics such as local limits of random walks, mixing/cutoff phenomena, or directed polymers. Some familiarity with measure-theoretic probability theory (as in MATH 5411) is helpful, but not strictly necessary to follow this course.