This monograph is a study of the local bifurcations of multiparameter symplectic maps of arbitrary dimension in the neighbourhood of a fixed point. The problem is reduced to a study of critical points of an equivariant gradient bifurcation problem, using the correspondence between orbits of a symplectic map and critical points of an action functional. New results on singularity theory for equivariant gradient bifurcation problems are obtained and then used to classify singularities of bifurcating period-q points. Of particular interest is the presentation of a general framework for analyzing group-theoretic aspects and singularities of symplectic maps (particularly period-q points). Topics include bifurcations when the symplectic map has spatial symmetry and a theory for the collision of multipliers near rational points with and without spatial symmetry. The monograph also includes 11 self-contained appendices each with a basic result on symplectic maps.