This review is organized such that exact Hamiltonians for matter coupled to a single mode are initially introduced, and the following three sections layer on approximations, going all the way to the semiclassical approximation. As such, Sect. II introduces different forms of the full Hilbert space Hamiltonian, derived from the fundamental Minimal Coupling Hamiltonian. Then, in Sect. III, the truncation of the full Hilbert space is considered, with discussions of the various methods for addressing the gauge ambiguities caused by such projections. In Sect. IV, the simplified quantum optics models are discussed and benchmarked relative to the truncated matter Hamiltonians. Then, Sect. V provides a brief comparison of cQED methods with Floquet theory, which is under the semiclassical approximation. Using insights from this path, Sect. VI extends the formalism to more generalized forms of cQED Hamiltonians for systems with many modes and many molecules. The future perspectives and analysis are provided in Sect. VII. Properties of a formaldehyde molecule inside a cavity, demonstrating more complex polariton eigenspectra and changes in transition densities due to the cavity.