We begin a rigorous, nonperturbative investigation of quantum field theories with local internal symmetries. We discuss the lattice approximation of Yang-Mills fields and of fermion fields in the Euclidean setup and we verify physical positivity for the Schwinger functions of these approximations. This implies the existence of a positive self-adjoint transfer matrix. We then prove existence and analyticity of the infinite volume limit of strongly coupled Yang-Mills theories on the lattice and we verify Wilson's confinement bound. Finally we present a rigorous treatment of the Higgs mechanism in lattice gauge theories.