  We introduce a parametrized notion of genericity for Delaunay triangulations
which, in particular, implies that the Delaunay simplices of $\delta$-generic
point sets are thick. Equipped with this notion, we study the stability of
Delaunay triangulations under perturbations of the metric and of the vertex
positions. We quantify the magnitude of the perturbations under which the
Delaunay triangulation remains unchanged.
