  We extend Schaeffer's bijection between rooted quadrangulations and
well-labeled trees to the general case of Eulerian planar maps with prescribed
face valences, to obtain a bijection with a new class of labeled trees, which
we call mobiles. Our bijection covers all the classes of maps previously
enumerated by either the two-matrix model used by physicists or by the
bijection with blossom trees used by combinatorists. Our bijection reduces the
enumeration of maps to that, much simpler, of mobiles and moreover keeps track
of the geodesic distance within the initial maps via the mobiles' labels.
Generating functions for mobiles are shown to obey systems of algebraic
recursion relations.
