  The phenomenology of supersymmetric models of inflation, where the
inflationary vacuum energy is dominated by D-terms of a U(1), is investigated.
Particular attention is paid to the questions of how to arrange for sufficient
e-folds of inflation to occur, what kind of thermal history is expected after
the end of inflation, and how to implement successful baryogenesis. Such models
are argued to require a more restrictive symmetry structure than previously
thought. In particular, it is non-trivial that the decays of the fields driving
D-inflation can reheat the universe in such a way as to avoid the strong
gravitino production constraints. We also show how the initial conditions for
Affleck-Dine baryogenesis can arise in these models and that the simplest flat
directions along which baryon number is generated can often be ruled out by the
constraints coming from decoherence of the condensate in a hot environment. At
the end, we find that successful reheating and baryogenesis can take place in a
large subset of D-inflationary models.
