  We study how higher-order gravity affects Higgs inflation in the Palatini
formulation. We first review the metric and Palatini formulations in
comparative manner and discuss their differences. Next cosmic inflation driven
by a scalar field and inflationary observables are discussed. After this we
review the Higgs inflation and compute the inflationary observables both in the
metric and Palatini formulations. We then consider adding higher-order terms of
the curvature to the action. We derive the equations of motion for the most
general action quadratic in the curvature that does not violate parity in both
the metric and Palatini formulations. Finally we present a new result. We
analyse Higgs inflation in the Palatini formulation with higher-order curvature
terms. We consider a simplified scenario where only terms constructed from the
symmetric part of the Ricci tensor are added to the action. This implies that
there are no new gravitational degrees of freedom, which makes the analysis
easier. As a new result we found out that the scalar perturbation spectrum is
unchanged, but the tensor perturbation spectrum is suppressed by the
higher-order curvature couplings.
