The course provides an introduction to the theory of multiscale methods, and the techniques of averaging and homogenization in particular. The theory will be exemplified by application to ordinary and stochastic differential equations, Markov chains and partial differential equations. The course will comprise 10 introductory lectures on averaging and homogenization by G.A. Pavliotis (Imperial, on partial differential equations) and by A.M. Stuart (Warwick, on stochastic processes). There will also be several guest lectures by experts in multiscale phenomena. These will cover applications and theory. The course is aimed primarily at graduate students from mathematics and statistics, but will also be of interest to students working in closely allied areas of engineering and physics. Furthermore, postdoctoral and young researchers are also encouraged to attend.