This thesis presents the development of a fully covariant approach to scalar-tensor theories of gravity in the context of inflation, as well as a covariant treatment of trajectory fine tuning in multifield models. Our main result is the introduction of frame covariance as a way to confront the frame problem in inflation. We treat the choice of a gravitational frame in which a theory is presented as a particular instance of gauge fixing. We take frame covariance beyond the tree level by virtue of the Vilkovisky-De Witt formalism, which was originally developed with the aim of removing the gauge and reparametrisation ambiguities from the path integral. Adopting an analogous approach, we incorporate conformal covariance to the Vilkovisky-De Witt formalism, demonstrating that the choice of a conformal frame is not physically important. This makes it possible to define a unique action even in the presence of matter couplings. We therefore show that even if the matter picture of the Universe may appear different in conformally related models, the underlying theory is independent of its frame representation. We further examine the relation between parameter fine tuning and initial condition fine tuning. Even though conceptually distinct, they both adversely affect the robustness of using established particle physics models to drive inflation. As a way to remedy this, we note that the presence of additional scalar degrees of freedom can "rescue'' particular models that are ruled out observationally by shifting the burden of fine tuning from the parameters to the choice of slow-roll trajectory. We refer to this uniquely multifield phenomenon as "trajectory fine tuning'', and we propose a method to quantify the sensitivity of multifield models to it. We illustrate by presenting examples of both single-field and multifield models of inflation, as well as F(R) and F(phi,R) theories.