Given a Noetherian local ring (R,m), one might ask whether there exists an R-module M and a system of parameters on R which form a regular sequence on M. Furthermore one might ask whether such M can be chosen to be finitely generated. Modules M satisfying these properties are called Big and Small Cohen-Macaulay modules, respectively. This talk gives a brief historical overview of progress on two major conjectures regarding the existence of Big and Small Cohen-Macaulay Modules.