  We give here a self contained and elementary introduction to the
Conley-Zehnder index for a path of symplectic matrices. We start from the
definition of the index as the degree of a map into the circle for a path
starting at the identity and ending at a matrix for which 1 is not an
eigenvalue. We prove some properties which characterize this index using normal
forms for symplectic matrices obtained from geometrical considerations. We
explore the relations to Robbin-Salamon index for paths of Lagrangians. We give
an axiomatic characterization of the generalization of the Conley-Zehnder index
for any continuous path of symplectic matrices defined by Robbin and Salamon.
