The talk provides an overview of the essential aspects of the extended metaplectic representation from the point of view of its geometric action on phase space. In our discussion we concentrate on reproducing formulae arising from connected Lie-subgroups of the semi-direct product of the Heisenberg group and the symplectic group on which the representation is defined. Wigner distribution provides a natural interpretation of the action on functions in terms of phase space geometry and it occurs to be an adequate tool for recognizing the fact that reproducing formulae correspond to phase space coverings. We illustrate our general statements with a list of representative examples.