  We present an introduction to cosmic inflation in the framework of Palatini
gravity, which provides an intriguing alternative to the conventional metric
formulation of gravity. In the latter, only the metric specifies the spacetime
geometry, whereas in the former, the metric and the spacetime connection are
independent variables-an option that can result in a gravity theory distinct
from the metric one. In scenarios where the field(s) responsible for cosmic
inflation are non-minimally coupled to gravity or the gravitational sector is
extended, assumptions about the underlying gravitational degrees of freedom can
have substantial implications for the observational effects of inflation. We
examine this explicitly by discussing various compelling scenarios, such as
Higgs inflation with non-minimal coupling to gravity, Higgs inflation with
non-minimal derivative coupling, $\mathcal{R}^2$ inflation, and beyond. We also
comment on reheating in these models. Finally, as an application of the general
results of Palatini $\mathcal{R}^2$ inflation, we review a model of successful
quintessential inflation, where a single scalar field acts initially as the
inflaton and then becomes dynamical dark energy, in agreement will all
experimental constraints.
