  The excessive production of gravitinos in the early universe destroys the
successful predictions of nucleosynthesis. The thermal generation of gravitinos
after inflation leads to the bound on the reheating temperature, T_{RH}< 10^9
GeV. However, it has been recently realized that the non-thermal generation of
gravitinos in the early universe can be extremely efficient and overcome the
thermal production by several orders of magnitude, leading to much tighter
constraints on the reheating temperature. In this paper, we first investigate
some aspects of the thermal production of gravitinos, taking into account that
in fact reheating is not instantaneous and inflation is likely to be followed
by a prolonged stage of coherent oscillations of the inflaton field. We then
proceed by further investigating the non-thermal generation of gravitinos,
providing the necessary tools to study this process in a generic time-dependent
background with any number of superfields. We also present the first numerical
results regarding the non-thermal generation of gravitinos in particular
supersymmetric models.
