This chapter lays out the foundation of gauge field theories, at least in part, and computes the one-loop renormalization of the gauge coupling. After a leisurely introduction to the gauge field strength and the action, in the geometric convention and anti-Hermitian basis, we introduce the effective action with an infrared cut-off. One-loop running of the QED coupling constant is computed via the heat kernel method, which inevitably leads to the Landau pole at the high-energy side, where the coupling diverges and beyond which the gauge kinetic term turns negative. We then turn to Yang-Mills theories, in a language suitable for classical Lie groups. The Faddeev-Popov gauge-fixing procedure and the background-field method are given a full treatment. Asymptotic freedom is derived, which is followed by applications to supersymmetric Yang-Mills theories and to spontaneously broken theories. Along the way, we turn to Wilsonian renormalization, in preparation for the next chapter, and emphasize how the Landau pole phenomenon of QED and the divergent Yang-Mills coupling at the low scale are fundamentally different, despite the apparent similarity.