  We introduce a new approach to modified gravity which generalizes the
recently proposed hybrid metric-Palatini gravity. The gravitational action is
taken to depend on a general function of both the metric and Palatini curvature
scalars. The dynamical equivalence with a non-minimally coupled bi-scalar field
gravitational theory is proved. The evolution of cosmological solutions is
studied using dynamical systems techniques.

General Relativity and Quantum Cosmology