In my talk, I will start by introducing the basics of Markov Chain Monte Carlo (MCMC) methods. I will start with the Metropolis and Gibbs samplers, and the proceed to the Hamiltonian Monte Carlo sampler. A key focus will be to describe the domains of applicability of each of these sampling methods and the difficulties they encounter when applied. I will then describe strategies to overcome some of the difficulties encountered with basic versions of these methods. I will also touch on output diagnostics, and determining when a sampler is working as desired. Finally, I will consider the case of sampling where the likelihood of a model is expensive to compute and how MCMC can be used in this situation. The latter case may be of interest in astronomy applications.