In Sec. 2, we discussed the rigorous theoretical background of molecular cavity QED. We first reviewed the basic theory of the molecular Hamiltonian (Sec. 2.1) and quantum electrodynamics (Sec. 2.2). Sec. 2.3 further reviews different forms of the QED Hamiltonians under different gauges and provides a clear connection among them through gauge transformations. Even though the theory of QED goes back to the mid-twentieth century, we discussed recent advances made in resolving gauge ambiguities to describe interactions between light and matter. In particular, Sec. 2.4 highlighted several possible causes of such ambiguities and their resolutions, which enable consistent physics regardless of the chosen gauge. In Sec. 2.5, we then connected the most rigorous QED Hamiltonian with various approximate Hamiltonians commonly used in the quantum optics community, which can be achieved through intuitive arguments and simple mathematical approximations of the rigorous Hamiltonian. Finally, in Sec. 2.6 we discussed lightmatter interactions between many molecules and many cavity modes inside a Fabry-Pérot cavity, which is one of the most experimentally relevant setups.