In this project we will give a comprehensive introduction to inverse problems and Markov chain Monte Carlo methods. We will begin by introducing inverse problems and how to solve them, focusing on the Bayesian inference approach. Then we will argue for the value in using Markov chain Monte Carlo methods when using this approach, thus shifting the focus of our project onto these methods. We will begin by first introducing Markov chains to ensure that the reader has the appropriate background to understand the theory behind these methods. Then, we will shift our attention to Markov chain Monte Carlo methods, in particular to the Metropolis Hastings algorithm. Subsequently, we will propose a different set of Markov chain Monte Carlo methods called non reversible samplers. These, as we shall see, have highly desirable properties. We hope to persuade the reader to use these methods by comparing the performances of these different algorithms in a series of illustrative examples. To conclude, we look will look at inverse problems in imaging.