The coupling method is a versatile proof technique in modern probability theory with a wide range of applications. Coupling consists of constructing copies of random elements on a common probability space in order to deduce properties or discover similarities or relations between them. In this master’s thesis, the notion of coupling is introduced with a few examples and methods of construction. We define the total variation norm and coupling inequalities, which are basic tools for comparing random variables. It is proved that there exists a maximal coupling which is in some sense the best coupling. Furthermore, we study the relation between coupling and convergence of random variables. We use the coupling method in the Poisson approximation. We prove that the sum of Bernoulli random variables is close to the Poisson distribution. Also, coupling is a key tool in the study of asymptotic properties of stochastic processes. The standard result is the Markov chain convergence theorem. We construct two variations of coupling called the Ornstein coupling and the epsilon-coupling of random walks to prove asymptotic results. The main goal is to give the coupling proof of Blackwell’s renewal theorem. Coupling is also useful in exact sampling. We describe the coupling from the past algorithm for exact sampling from a finite set. In this method, coupled Markov chains are started from the points in the more and more distant past and run until they coalesce in the present. Thus, the algorithm determines when to stop and returns value distributed exactly according to the stationary distribution. Furthermore, we introduce the Ising model and describe its application to image analysis. Finally, we implement the algorithm on the Ising model and show the simulation results.