  An alternative for the construction of fundamental theories is the
introduction of Galileons. These are fields whose action leads to non higher
than second-order equations of motion. As this is a necessary but not
sufficient condition to make the Hamiltonian bounded from below, as long as the
action is not degenerate, the Galileon construction is a way to avoid
pathologies both at the classical and quantum levels. Galileon actions are,
therefore, of great interest in many branches of physics, specially in high
energy physics and cosmology. This proceedings contribution presents the
generalities of the construction of both scalar and vector Galileons following
two different but complimentary routes.
