In the past 15 years the potential theory of Markov chains with countable state space has been energetically developed by numerous authors. In particular the boundary theory of Markov chains first investigated by J.L. Doob, W. Feller and G.A. Hunt receives much current attention. These lectures aim to present the basic ideas of this theory to graduate students who had previous exposure to basic probability theory. The lectures were given with special emphasis on the probabilistic basis and significance of potential theory. Since boundary theory was one of the main objectives, more attention was given to transient chains than to recurrent chains. Sections 1 through 4 present the basic concepts of Markov chains. Section 5 is a brief treatment of recurrent chains, Sections 7 through 9 form an introduction to potential theory and Section 10 is an introduction to the Martin boundary including a discussion of the boundaries for random walks and the Polya urn scheme. The lecture notes do not form a complete exposition of the theory but are meant as an introduction to more detailed treatments such as is found in the book of Kemeny, Snell, and Knapp and to the current literature. I would like to thank the Canadian Mathematical Congress and in particular Dr. John McNamee and Professor Ron Руке for their encouragement and assistance with the presentation of the lectures and the preparation of these notes.