The class of normal maps was recently investigated by Robinson and Ralph in connection with the study of a variational inequality defined on a polyhedral set. In this paper a generalization of such a map is considered, and the associated generalized normal equation is studied. The latter provides a unified formulation of several generalized variational inequality and complementarity problems. Using degree theory, some sufficient conditions for the existence of a zero of a generalized normal map are established and the stability of a generalized normal equation at a solution is analyzed. Specializations of the results to various applications are discussed.