  We define Maslov--type indices associated to contact and symplectic
transformation groups. There are two such families of indices. The first class
of indices are maps from the homotopy groups of the contactomorphism or
symplectomorphism group to a quotient of the integers. These are based on a
generalization of the Maslov index. The second class of indices are maps from
the homotopy groups of the space of contact structures or the space of
cohomologous symplectic forms to the homotopy groups of a simple homogeneous
space. We provide a detailed construction and describe some properties of these
indices and their applications.
