Strong interactions between molecular electronic and photonic degrees of freedom (i.e., electronic strong and ultra-strong coupling) can fundamentally alter chemical structure, reactivity, and phenomenology. Predictive theoretical and computational models for molecules under electronic strong coupling must capture the quantum nature of electronic and photonic degrees of freedom. In the limit of molecular electronic strong or ultra-strong coupling to one or a few molecules, it is desirable to treat the molecular electronic degrees of freedom using the tools of ab initio quantum chemistry, yielding an approach referred to as ab initio cavity quantum electrodynamics (ai-QED), where the photon degrees of freedom are treated at the level of cavity quantum electrodynamics. Two complementary approaches have emerged for ai-QED: (1) parameterized CQED (pQED), a twostep approach where the matter degrees of freedom are computed using existing electronic structure theories, enabling one to build rigorous ai-QED Hamiltonians in a basis of many-electron eigenstates, and (2) self-consistent CQED (scQED), a one-step approach where electronic structure methods are generalized to include coupling between electrons and photon degrees of freedom.