  We review the different frameworks in which Galileon scalar fields have been
seen to emerge such an in DGP, New Massive Gravity and Ghost-free massive
Gravity and emphasize their relation with the Lovelock invariant in braneworld
models. The existence of a non-renormalization theorem for Galileon scalar
fields makes them especially attractive candidates for inflation as well as for
late-time acceleration. In particular we review the self-accelerating and
degravitating branches of solutions present in Galileon models when arising
from Massive Gravity and discuss their phenomenology.

Cosmology and Nongalactic Astrophysics;General Relativity and Quantum Cosmology;High Energy Physics - Theory