The aim of this paper is to study the limiting behavior of Markov chains. We begin by introducing the fundamental definitions and theorems of Markov chain theory. A theorem which states that an irreducible and aperiodic Markov chain approaches its stationary distribution motivates us to study the speed of that convergence. Therefore, we define the total variation norm as a measure of distance between two probability distributions, and mixing times as the number of steps needed for a Markov chain to come close to the stationary distribution. Applying the theory of linear operators on a reversible transition matrix within a appropriately defined unitary space allows us to prove that the matrix is self-adjoint and so we construct an orthonormal basis which will be useful in upcoming proofs. Using eigenvalues of the transition matrix, we define the (absolute) spectral gap and relaxation time of a Markov chain and show how those values are used to establish upper and lower bounds on mixing time. We also discuss two important consequences of the mixing time theory. The first example is a problem that sparked the development of the mixing time theory in the first place – the optimal number of shuffles for a deck of cards. The second application is a generalization of the law of large numbers and Chebyshev inequality to Markov chains, which is strongly related to the theory MCMC methods are built upon.