This chapter develops the scalar and vector electromagnetic potentials as the fundamental dynamical variables for coupling fields to charged matter. Starting from Maxwell’s equations, it introduces definitions of φ and A, shows how to obtain the electric and magnetic fields from these potential fields, and explores the gauge freedom. Coulomb and Lorenz gauges are derived and connected to retarded solutions. The chapter then recasts particle dynamics in the Lagrangian and Hamiltonian formalisms using a velocity-dependent potential, U = qϕ – q v۰A, deriving the Lorentz force from the Euler–Lagrange equations and the canonical momentum, p = Mv + qA. Finally, it carries this structure into quantum mechanics: minimal coupling in the Schrödinger equation, gauge invariance of observables, and the description of motion via propagators and Feynman’s path integral, where the phase is determined by the action built from the same electromagnetic potentials.