  We review the new theory of modified supergravity, dubbed F(\cal R)
supergravity, and some of its recent applications to inflation and reheating in
the early universe cosmology. The F(\cal R) supergravity is the N=1 locally
supersymmetric extension of the f(R) gravity in four space-time dimensions. A
manifestly supersymmetric formulation of the F(\cal R) supergravity exist in
terms of N=1 superfields, by using the (old) minimal Poincar'e supergravity in
curved superspace. We find the conditions for stability, the absence of ghosts
and tachyons. Three models of the F(\cal R) supergravity are studied. The first
example is devoted to a recovery of the standard (pure) N=1 supergravity with a
negative cosmological constant from the F(\cal R) supergravity. As the second
example, a generic {\cal R}^2-type supergravity is investigated, and the
existence of the AdS bound on the scalar curvature is found. As the third (and
most important) example, a simple viable realization of chaotic inflation in
supergravity is found. Our approach is {\it minimalistic} since it does not
introduce new exotic fields or new interactions, beyond those already present
in (super)gravity. The universal reheating mechanism is automatic. We establish
the consistency of our approach and also apply it to preheating and reheating
after inflation. The Higgs inflation and its correspondence to the Starobinsky
inflation is established in the context of supergravity. We briefly review
other relevant issues such as non-Gaussianity, CP-violation, origin of baryonic
asymmetry, lepto- and baryo-genesis. The F(\cal R) supergravity has promise for
possible solutions to those outstanding problems too.
