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Recent work incorporating geometric ideas in Markov chain Monte Carlo isreviewed in order to highlight these advances and their possible application ina range of domains beyond Statistics. A full exposition of Markov chains andtheir use in Monte Carlo simulation for Statistical inference and moleculardynamics is provided, with particular emphasis on methods based on Langevindiffusions. After this geometric concepts in Markov chain Monte Carlo areintroduced. A full derivation of the Langevin diffusion on a Riemannianmanifold is given, together with a discussion of appropriate Riemannian metricchoice for different problems. A survey of applications is provided, and someopen questions are discussed.

Information-geometric Markov Chain Monte Carlo methods using Diffusions