  In the present thesis, using an effective field theory point of view, we
explore theories of single-field inflation where higher derivative operators
become relevant, affecting in a novel way the dynamics and therefore the
observations. For instance, concerning the scalar spectrum, they allow for
measurable equilateral non-Gaussianity, whose amplitude can differ
significantly from the predictions of other existing models. Moreover, we show
that the stability and the consistency of such theories are ensured by an
approximate Galileon symmetry. Indeed, being generically possible to build an
invariant theory under Galileon transformations in flat space-time, it is
instead well known that such a symmetry is unavoidably broken by gravity. In
principle, this might ruin the nice and interesting properties of the Galileons
in flat backgrounds, such as the non-renormalization theorem. However, we find
that this does not happen if the Galileon invariance is broken only weakly, in
a well defined sense, by a suitable coupling to gravity, providing therefore an
extension of the quantum non-renormalization properties in curved space-times.
Hence, besides discussing the phenomenological consequences and the
observational predictions for inflation, we apply such Galileon theories to the
context of the late-time acceleration of the Universe. In the last part, in
order to probe non-standard primordial scenarios, they are also employed in a
cosmology where the Big Bang singularity is smoothed down and the Universe
emerges from a Minkowski space-time, in a well defined extension at all times
of the Galilean Genesis scenario.
