The study of Cohen-Macaulay rings and modules has been a central topic in Commutative Algebra for many years. Among other things, they have played a major role in the investigations of the "Homological Conjectures", a set of problems on finite projective dimension, intersection theory, and related subjects. More recently, as a result of advances in the homological conjectures and developments in Arithmetic Geometry, a number of questions have come up about "almost" Cohen-Macaulay modules and algebras. In this talk I will give some background to these topics, discuss what almost Cohen-Macaulay algebras are and why they are interesting, and present various recent developments and open problems in this area.