Maximal Cohen-Macaulay modules over surface singularities

This is a survey article about properties of Cohen-Macaulay modules over surface singularities. We discuss various results on the Macaulayfication functor, reflexive modules over simple, quotient and minimally elliptic singularities, geometric and algebraic McKay Correspondence. Finally, we describe matrix factorizations corresponding to the indecomposable Cohen-Macaulay modules over the non-isolated singularities $A_\infty$ and $D_\infty$.