  In this work, we study cosmological and astrophysical applications of the
recently proposed generalized hybrid metric-Palatini gravity theory, which
combines features of both the metric and the Palatini approaches to the
variational method in $f\left(R\right)$ gravity. This theory arises as a
natural generalization of the hybrid metric-Palatini gravity which has been
proven to be the first theory to unify the cosmic acceleration with the solar
system constraints, without resource to the chameleon mechanism.
  In the cosmological point of view, we show using reconstruction methods that
the usual power-law and exponential scale factor behaviors in FLRW universes
exist for various different distributions of matter, along with solutions for
collapsing universes. Using the dynamical system approach, we also show that no
global attractors can exist in the cosmological phase space and that stable
universes can either be described by scale factors that diverge in finite time
or asymptotically approach constant values. Furthermore, we also study the
cosmological phase space of theories of gravity with terms of order six and
eight in the derivatives of the metric and we conclude that the higher-order
terms are not neglectable.
  In the area of astrophysics, we show that using the junction conditions of
the theory it is possible to obtain solutions for compact objects supported by
thin-shells, such as self-gravitating shells with and without perfect fluids on
their exteriors, and also traversable wormhole solutions which satisfy the null
energy condition for the whole spacetime, thus not needing the support of
exotic matter. Furthermore, we show that there exist specific forms of the
action for which the massive scalar degree of freedom of the theory is stable
in the scope of rotating black-holes described by the Kerr metric.
