Non-Local Gravity Cosmology: an Overview
  We discuss some main aspects of theories of gravity containing non-local
terms in view of cosmological applications. In particular, we consider various
extensions of General Relativity based on geometrical invariants as $f(R,
\Box^{-1} R)$, $f({\cal G}, \Box^{-1} {\cal G})$ and $f(T, \Box^{-1} T)$
gravity where $R$ is the Ricci curvature scalar, $\cal G$ is the Gauss-Bonnet
topological invariant, $T$ the torsion scalar and the operator $\Box^{-1}$
gives rise to non-locality. After selecting their functional form by using
Noether Symmetries, we find out exact solutions in a cosmological background.
It is possible to reduce the dynamics of selected models and to find analytic
solutions for the equations of motion. As a general feature of the approach, it
is possible to address the accelerated expansion of the Hubble flow at various
epochs, in particular the dark energy issues, by taking into account
non-locality corrections to the gravitational Lagrangian. On the other hand, it
is possible to search for gravitational non-local effects also at astrophysical
scales. In this perspective, we search for symmetries of $f(R, \Box^{-1} R)$
gravity also in a spherically symmetric background and constrain the free
parameters, Specifically, by taking into account the S2 star orbiting around
the Galactic Centre SgrA$^*$, it is possible to study how non-locality affects
stellar orbits around such a massive self-gravitating object.
