In this work, we obtained characterization curves of constant breadth in four dimensional Galilean space in terms of Frenet-Serret vector fields. First we investigate an explicit characterization of curves of constant breadth of four dimensional Galilean space.In addition,it has been observed that differential equation of 4th order of a constant breadth curve in four dimensional Galilean space.Finally,some differential and integral characterizations of the mentioned curves are expressed by the classical differential geometry methods.