  In this work, we review a plethora of modified theories of gravity with
generalized curvature-matter couplings. The explicit nonminimal couplings, for
instance, 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, implying non-geodesic motion and consequently leads
to the appearance of an extra force. Applied to the cosmological context, these
curvature-matter couplings lead to interesting phenomenology, where one can
obtain a unified description of the cosmological epochs. We also consider the
possibility that the behavior of the galactic flat rotation curves can be
explained in the framework of the curvature-matter coupling models, where the
extra-terms in the gravitational field equations modify the equations of motion
of test particles, and induce a supplementary gravitational interaction. In
addition to this, these models are extremely useful for describing dark
energy-dark matter interactions, and for explaining the late-time cosmic
acceleration.
