A nodal curve is determined by its normalization, marked with the
points lying over its nodes. The same holds for a curve with ordinary
cusps. But to specify a curve with other singularities, extra
"crimping data" is necessary, beyond the isomorphism types of the
singularities. We explain this phenomenon, construct a moduli space of
crimping data via deformation theory in characteristic zero and in
positive characteristic, and give a structure theorem. Then we show
how to compactify the space in certain cases and see what this tells
us about possible modular compactifications of the moduli space of
smooth curves of a given genus, as studied for example in work of
Hassett and Smyth.