Before directly discussing various exact cavity QED Hamiltonians, we define the formalism used in the rest of the review for the two primary degrees of freedom (DOF) covered in this review, namely the molecular Hamiltonian and the photonic Hamiltonian. Following this brief overview of these independent Hamiltonians, the rest of this section focuses on different representations of QED Hamiltonians that describe light-matter interactions: the minimal coupling Hamiltonian (Sect. II A), the dipole gauge Hamiltonian (Sect. II B), and the assymptotically decoupled Hamiltonian (Sect. II C).