  In the present work, we review some general aspects of modified gravity
theories, investigating mathematical and physical properties and, more
specifically, the feature of viable and realistic models able to reproduce the
dark energy epoch and the early-time inflation. We will discuss the black hole
solutions in generalized theories of gravity: it is of fundamental interest to
understand how properties and laws of black holes in General Relativity can be
addressed in the framework of modified theories. In particular, we will discuss
the energy issue and the possibility to derive the First Law of thermodynamics
from the field equations. Then, in the analysis of cosmological solutions, we
will pay a particular attention to the occurrence of finite-time future
singularities and to the possibility to avoid them in $\mathcal
F(R,G)$-gravity. Furthermore, realistic models of $F(R)$-gravity will be
analyzed in the detail. A general feature occurring in matter era will be
shown, namely the high derivatives of Hubble parameter may be influenced by the
high frequency oscillation of the dark energy and some correction term is
required in order to stabilize the theory at high redshift. The inflationary
scenario is also carefully analyzed and an unified description of the universe
is risen. In the final part of the work, we will have a look at the last
developments in modified gravity, namely we will investigate cosmological and
black hole solutions in a covariant field theory of gravity and we will
introduce the extended "teleparallel" $F(T)$-gravity theories. A nice
application to the dark matter problem will be presented.
