Most of the time, the definitions of minima, saddle points or more generally order p (p=0,...,n) critical points, do not mention the possibility of having zero Hessian eigenvalues. This feature reflects some flatness of the potential energy hypersurface in a special eigendirection which is not often taken into account. Thus, the definitions of critical points are revisited in a more general framework within this context. The concepts of bifurcation points, branching points, and valley ridge inflection points are investigated. New definitions based on the mathematical formulation of the reaction path are given and some of their properties are outlined. (C) 2000 American Institute of Physics. [S0021-9606(00)01110-7].