Cohen-Macaulay rings are a well-studied class of rings that have been used in many areas of math, including algebraic geometry and combinatorics. In this talk, we introduce Cohen-Macaulay rings and some of their properties. Along the way, we will see some examples, including rings defined via ideals generated by determinants of certain matrices. If time permits, we will outline how Cohen-Macaulayness and determinantal ideals can be applied in the field of algebraic statistics to study varieties defined by moments of statistical distributions.