After reviewing the motivations for cosmological inflation formulated in the formalism of supersymmetry, we argue that the appropriate framework is that of no-scale supergravity. We then show how to construct within this framework inflationary models whose predictions for the tilt in the spectrum of scalar perturbations, $n_s$, and the ratio, $r$, of tensor and scalar perturbations coincide with those of the $R+R^2$ model of inflation proposed by Starobinsky. A more detailed study of no-scale supergravity reveals a structure that is closely related to that of $R^2$ modifications of the minimal Einstein-Hilbert action for general relativity, opening avenues for constructing no-scale de Sitter and anti-de Sitter models by combining pairs of Minkowski models, as well as generalizations of the original no-scale Starobinsky models of inflation. We then discuss the phenomenology of no-scale models of inflation, including inflaton decay and reheating, and then the construction of explicit scenarios based on SU(5), SO(10) and string-motivated flipped SU(5)$\times$U(1) GUT models. The latter provides a possible model of almost everything below the Planck scale, including neutrino masses and oscillations, the cosmological baryon asymmetry and cold dark matter, as well as $n_s$ and $r$.