  The theory of General Relativity was established on a spacetime manifold
equipped with a metric tensor, $(\mathcal{M}_4,\text{g})$, and the connection
on $\mathcal{M}_4$ identified with the Levi-Civita one. Even though there are
valid reasons to assume a torsionless manifold that preserves the metric, it
was shown that dealing away with these assumptions the Levi-Civita condition
can be reproduced at the level of equations of motion of GR. It was not long
before the equivalence of General Relativity between the two descriptions,
known as the Palatini or first-order formalism in which the connection is
independent of the metric, and the conventional metric or second-order
formalism, was broken for more complicated actions involving higher-order
curvature invariants and/or nonminimal couplings between the gravitational and
matter sector. Nowadays these types of theories are prominent in modeling
inflation where they have found major success. Since the paradigm of inflation
is fused with the gravitational degrees of freedom and thus their
parametrisation, it is interesting to understand how the predictions of these
models differ between the two formulations. One of the outstanding models of
inflation is the Starobinsky or quadratic gravity model. However in the
Palatini formalism the scalar degree of freedom sourced by the $R^2$ term is
nondynamical and is unable to drive an inflationary phase. In order for
inflation to be realised in the first-order formalism the Starobinsky model has
to be coupled with a fundamental scalar field that will assume the role of the
inflaton. In this thesis we investigate different inflationary scenarios,
starting with previously ruled-out models, where we find that the $R^2$ term
has a significant role in flattening the Einstein-frame inflaton potential and
thus giving the opportunity for these models to come in contact with
observations.
