  Recently, the phenomenology of f(R) gravity has been scrutinized motivated by
the possibility to account for the self-accelerated cosmic expansion without
invoking dark energy sources. Besides, this kind of modified gravity is capable
of addressing the dynamics of several self-gravitating systems alternatively to
the presence of dark matter. It has been established that both metric and
Palatini versions of these theories have interesting features but also manifest
severe and different downsides. A hybrid combination of theories, containing
elements from both these two formalisms, turns out to be also very successful
accounting for the observed phenomenology and is able to avoid some drawbacks
of the original approaches. This article reviews the formulation of this hybrid
metric-Palatini approach and its main achievements in passing the local tests
and in applications to astrophysical and cosmological scenarios, where it
provides a unified approach to the problems of dark energy and dark matter.
