Cohen-Macaulay Nilpotent Schemes

The first part of this paper, written mainly for nonspecialists, is a short and partial survey about the construction and classification of nilpotent Cohen-Macaulay scheme structures on a scheme "less nilpotent"(e.g. a smooth variety) as support. The second part presents some old and new examples about degree 4 curves in the projective space and inexistence of certain vector bundles.