  Mean-field models, while they can be cast into an {\it extensive}
thermodynamic formalism, are inherently {\it non additive}. This is the basic
feature which leads to {\it ensemble inequivalence} in these models. In this
paper we study the global phase diagram of the infinite range
Blume-Emery-Griffiths model both in the {\it canonical} and in the {\it
microcanonical} ensembles. The microcanonical solution is obtained both by
direct state counting and by the application of large deviation theory. The
canonical phase diagram has first order and continuous transition lines
separated by a tricritical point. We find that below the tricritical point,
when the canonical transition is first order, the phase diagrams of the two
ensembles disagree. In this region the microcanonical ensemble exhibits energy
ranges with negative specific heat and temperature jumps at transition
energies. These two features are discussed in a general context and the
appropriate Maxwell constructions are introduced. Some preliminary extensions
of these results to weakly decaying nonintegrable interactions are presented.
