Despite being a principle of classical field theory, gauge invariance of electrodynamics revealed its deep significance and found its far-reaching interpretation only in relation to quantum mechanics of electrons and the Schrödinger equation. In this chapter, we study the generalization of the concept of a locally invariant gauge theory to non-Abelian gauge groups constructed by following the model of Maxwell theory. This generalization may seem a little academic at first glance because, besides the Maxwell field, it contains further massless gauge fields which are unknown to macroscopic physics. However, it becomes physically realistic if it is combined with the phenomenon of spontaneous symmetry breaking. Both concepts, non-Abelian gauge theory and spontaneous symmetry breaking, initially are purely classical concepts. At the same time, one lays the (classical) foundations for the gauge theories of the fundamental interactions which nowadays are generally accepted and whose validity has been confirmed by numerous experiments. This chapter describes the foundations for the construction of such a theory, within a classical (i.e. nonquantum) framework. Only when introducing fermionic particles (such as quarks and leptons) does the quantization of gauge theories become mandatory.