Maximal Cohen Macaulay modules are studied in homological algebra and singularity theory. They tell us a lot about the structure of a commutative ring, but in general it is hard to write down such a module explicitly. However, for hypersurface rings, Eisenbud's matrix factorizations provide a good framework to work with them. Motivated by Physics, Maximal Cohen Macaulay modules have been established as a useful tool in construction of noncommutative desingularizations and recently also have been studied in the context of representation theory of algebras, in particular cluster structures on categories of MCM modules. These lectures should serve as an overview of the field to acquaint the participants with important results and techniques, with a particular focus on motivating examples like Auslander's algebraic version of the McKay correspondence.