MAA305 presents the basic theory of discrete Markov chains. It starts by introducing the Markov property and then moves on to develop fundamental tools such as transition matrices, recurrence classes and stopping times. With these tools at hand, the strong Markov property is proven and applied to the study of hitting probabilities. In the second part of the course, we introduce the notion of stationary distribution and prove some basic existence and uniqueness results. Finally, the long time behavior of Markov chains is investigated by proving the ergodic Theorem and exponential convergence under Doblin's condition. The course is concluded by surveying some stochastic algorithm whose implementation relies on the construction of an appropriate Markov Chain, such as the Metropolis-Hastings algorithm.