  Many extensions of Einstein's theory of gravity have been studied and
proposed with various motivations like the quest for a quantum theory of
gravity to extensions of anomalies in observations at the solar system,
galactic and cosmological scales. These extensions include adding higher powers
of Ricci curvature $R$, coupling the Ricci curvature with scalar fields and
generalized functions of $R$. In addition when viewed from the perspective of
Supergravity (SUGRA) many of these theories may originate from the same SUGRA
theory interpreted in different frames. SUGRA therefore serves as a good
framework for organizing and generalizing theories of gravity beyond General
Relativity. All these theories when applied to inflation (a rapid expansion of
early Universe in which primordial gravitational waves might be generated and
might still be detectable by the imprint they left or by the ripples that
persist today) can have distinct signatures in the Cosmic Microwave Background
radiation temperature and polarization anisotropies. In this review we give a
detailed discussion on the standard model of cosmology ($\Lambda$CDM),
inflation and cosmological perturbation theory. We survey the theories of
gravity beyond Einstein's General Relativity, specially which arise from SUGRA,
and study the consequences of these theories in the context of inflation and
put bounds on the theories and the parameters therein from the observational
experiments like Planck, Keck/BICEP. The possibility of testing these theories
in the near future in CMB observations and new data coming from colliders like
the LHC, provides an unique opportunity for constructing verifiable models of
particle physics and General Relativity.
