We present an introduction to cosmic inflation in the context of Palatini gravity, which is an interesting alternative to the usual metric theory of gravity. In the latter case only the metric $g_{mu u}$ determines the geometry of space-time, whereas in the former case both the metric and the space-time connection $Gamma^lambda_{mu u}$ are a priori independent variables - a choice which can lead to a theory of gravity different from the metric one. In scenarios where the field(s) responsible for cosmic inflation are coupled non-minimally to gravity or the gravitational sector is otherwise extended, assumptions of the underlying gravitational degrees of freedom can have a big impact on the observational consequences of inflation. We demonstrate this explicitly by reviewing several interesting and well-motivated scenarios including Higgs inflation, $R^2$ inflation, and $xi$-attractor models. We also discuss some prospects for future research and argue why $r=10^{-3}$ is a particularly important goal for future missions that search for signatures of primordial gravitational waves.
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