  We review our recent results on dark matter from Starobinsky supergravity. In
this context, a natural candidate for Cold Dark Matter is the gravitino. On the
other hand, assuming the supersymmetry broken at scales much higher than the
electroweak scale, gravitinos are super heavy particles. In this case, they may
be non-thermally produced during inflation, in turn originated by the scalaron
field with Starobinsky's potential.Assuming gravitinos as Lightest
supersymmetric particles (LSSP), the non-thermal production naturally accounts
for the right amount of cold dark matter. Metastability of the gravitino LSSP
leads to observable effects of their decay, putting constraints on the
corresponding Unstable or Decaying Dark Matters scenarios. In this model, the
gravitino mass is controlled by the inflaton field and it runs with it. This
implies that a continuous spectrum of superheavy gravitinos is produced during
the slow-roll epoch. Implications in phenomenology, model building in GUT
scenarios, intersecting D-branes models and instantons in string theories are
discussed.
