  We explore the paradigm according to which inflation is driven by a
four-dimensional strongly coupled dynamics coupled non-minimally to gravity. We
start by introducing the general setup, both in the metric and Palatini
formulation, for generic models of composite inflation. We then analyze the
relevant example where the inflaton is identified with the glueball field of a
pure Yang-Mills theory. We introduce the dilatonic-like glueball action which
is obtained by requiring saturation of the underlying Yang-Mills trace anomaly
at the effective action level. We couple the resulting action non-minimally to
gravity. We demonstrate that it is possible to achieve successful inflation
with the confining scale of the underlying Yang-Mills theory naturally of the
order of the grand unified energy scale. We also argue that within the metric
formulation models of composite inflation lead to a more consistent picture
than within the Palatini one. Finally we show that, in the metric formulation,
the model nicely respects tree-level unitarity for the scattering of the
inflaton field all the way to the Planck scale.
