A point P on a smooth hypersurface X of degree d in P-N is called a star point if and only if the intersection of X with the embedded tangent space T-P(X) is a cone with vertex P. This notion is a generalization of total inflection points on plane curves and Eckardt points on smooth cubic surfaces in P-3. We generalize results on the configuration space of total inflection points on plane curves to star points. We give a detailed description of the configuration space for hypersurfaces with two or three star points. We investigate collinear star points and we prove that the number of star points on a smooth hypersurface is finite. (C) 2009 Elsevier Inc. All rights reserved.