  We construct a consistent model of Galileon scalar electrodynamics. The model
satisfies three essential requirements: (1) The action contains higher-order
derivative terms, and obey the Galilean symmetry, (2) Equations of motion also
satisfy Galilean symmetry and contain only up to second-order derivative terms
in the matter fields and, hence do not suffer from instability, and (3) local
U(1) gauge invariance is preserved. We show that the non-minimal coupling terms
in our model are different from that of the real scalar Galileon models;
however, they match with the Galileon real scalar field action. We show that
the model can lead to an accelerated expansion in the early Universe. We
discuss the implications of the model for cosmological inflation.
