Given a finite group G, a normal subgroup N of G, and given an irreducible module of some subgroup of G that contains N, Clifford Theory describes the behavior of a large collection of modules for all the subgroups of G that contain N. In the classical case, i.e. in the case when the field is the complex numbers, the whole behavior is controlled by a transitive permutation representation of G/N and a single element of the group H^2(I/N, C^*), where I is the inertia group. The Brauer-Clifford Group unifies and generalizes these ideas to the case of arbitrary fields, i.e. including non algebraically closed fields of arbitrary characteristic. We discuss the definition of the Brauer-Clifford Group. We give some of its basic properties and mention some applications.