  We generalize the Galileon duality to any single scalar field Lagrangian
coupled locally to any matter field. Under the duality, a generalized Galileon
maps into another generalized Galileon via a one parameter group of
transformations, with only a simple modification of the Lagrangian functions.
We find a special class of generalized Galileons for which the duality is a
symmetry of the action. We further extend the duality to the case of vector
fields and give the dual formulations of the Maxwell and Proca theories. We
include arbitrary local couplings to matter fields and show that the duality
always maps a local interacting theory into a local interacting theory. We also
discuss the coupling to gravity and uncover a new class of Lorentz invariant
massive theories which map into themselves under the duality. Finally, we show
that the duality can be used to map solutions of a theory with superluminal
(luminal) group velocity into one with luminal (subluminal) group velocity. We
find that the duality nevertheless preserves the classical causal structure and
emphasize the need to include the quantum corrections to ascertain relativistic
causality.
