  We define and study the notion of a minimal Cohen-Macaulay simplicial
complex. We prove that any Cohen-Macaulay complex is shelled over a minimal one
in our sense, and we give sufficient conditions for a complex to be minimal
Cohen-Macaulay. We show that many interesting examples of Cohen-Macaulay
complexes in combinatorics are minimal, including Rudin's ball, Ziegler's ball,
the dunce hat, and recently discovered non-partitionable Cohen-Macaulay
complexes. We further provide various ways to construct such complexes.
