Information geometry and Markov chains are two powerful tools used in modern fields such as finance, physics, computer science, and epidemiology. In this survey, we explore their intersection, focusing on the theoretical framework. We attempt to provide a self-contained treatment of the foundations without requiring a solid background in differential geometry. We present the core concepts of information geometry of Markov chains, including information projections and the pivotal information geometric construction of Nagaoka. We then delve into recent advances in the field, such as geometric structures arising from time reversibility, lumpability of Markov chains, or tree models. Finally, we highlight practical applications of this framework, such as parameter estimation, hypothesis testing, large deviation theory, and the maximum entropy principle.