One-parametric mathematical programs with complementarity constraints are considered. The structure of the set of generalized critical points is analysed for the generic case. It is shown how this analysis can locally be reduced to the study of appropriate standard one-parametric finite problems. By applying the genericity result of the five types of Jongen, Jonker and Twilt for standard finite programs, we obtain a genericity result of the five types for one-parametric complementarity constrained problems. However, some effects differ from the situation in standard finite programming. The present investigations give the basis for path-following methods for solving one-parametric mathematical programs with complementarity constraints to be developed in the future.