Summary: This is an expository paper that presents various ideas related to nonasymptotic rates of convergence for Markov chains. Such rates are of great importance for stochastic algorithms that are widely used in statistics and in computer science. They also have applications to analysis of card shuffling and other areas. We attempt to describe various mathematical techniques that have been used to bound such rates of convergence. In particular, we describe eigenvalue analysis, random walks on groups, coupling, and minorization conditions. Connections are made to modern areas of research wherever possible. Elements of linear algebra, probability theory, group theory, and measure theory are used, but efforts are made to keep the presentation elementary and accessible.