  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 $D_\infty$.
