This research monograph explores new frontiers in Markov chains. Although time-homogeneous Markov chains are well understood, this is not at all the case with time-inhomogeneous ones. The book, after a review on the classical theory of homogeneous chains, including the electrical network approach, introduces several new models which involve inhomogeneous chains as well as related new types of random walks (for example, "coin turning", "conservative" and "Rademacher" walk). Scaling limits, the breakdown of the classical limit theorems as well as recurrence and transience are investigated. The relationship with urn models is the subject of two chapters, providing additional connections to other parts of probability theory. Random walks on random graphs are discussed as well, as an area where the method of electric networks is especially useful. This is illustrated by presenting random walks in random environments and random labyrinths. The monograph puts emphasis on showing examples and open problems besides providing rigorous analysis of the models. Several figures illustrate the main ideas, and a large number of exercises challenge the interested reader.