  Einstein's General Relativity (GR) is possibly one of the greatest
intellectual achievements ever conceived by the human mind. In fact, over the
last century, GR has proven to be an extremely successful theory, with a well
established experimental footing. However, the discovery of the late-time
cosmic acceleration, which represents a new imbalance in the governing
gravitational field equations, has forced theorists and experimentalists to
question whether GR is the correct relativistic theory of gravitation, and has
spurred much research in modified gravity, where extensions of the
Hilbert-Einstein action describe the gravitational field. In this review, we
perform a detailed theoretical and phenomenological analysis of two largely
explored extensions of $f(R)$ gravity, namely: (i) the hybrid metric-Palatini
theory; (ii) and modified gravity with curvature-matter couplings. Relative to
the former, it has been established that both metric and Palatini versions of
$f(R)$ gravity possess interesting features but also manifest severe drawbacks.
A hybrid combination, containing elements from both of these formalisms, turns
out to be very successful in accounting for the observed phenomenology and
avoids some drawbacks of the original approaches. Relative to the
curvature-matter coupling theories, these offer interesting extensions of
$f(R)$ gravity, where the explicit nonminimal couplings between an arbitrary
function of the scalar curvature $R$ and the Lagrangian density of matter,
induces a non-vanishing covariant derivative of the energy-momentum tensor. We
extensively explore both theories in a plethora of applications, namely, the
weak-field limit, galactic and extragalactic dynamics, cosmology, stellar-type
compact objects, irreversible matter creation processes and the quantum
cosmology of a specific curvature-matter coupling theory.
