Investigating the cavity QED computationally always requires a truncation of electronic states applied to the QED Hamiltonians , as the electronic Hilbert space in principle has an infinite basis size for any real system. Additionally, as these matter electronic states are often difficult to obtain, one typically projects the QED Hamiltonian to a few physically relevant electronic states, which can still produce accurate results in the low energy regime. However, when dealing with strong lightmatter coupling, the manner in which the truncation of the Hilbert space is performed drastically changes the accuracy and consequentially the convergence of results with basis size. As such, this section reviews the recent literature results on truncating the Hilbert space of these QED systems. Sects. III A and III B discuss two different ways to truncate the matter DOF of the dipole and Coulomb gauge Hamiltonians from Sects. II A and II B. Then, Sect. III C discusses a new representation of the light-matter coupling in a truncated Hilbert space, the polarized Fock state (PFS) Hamiltonian. Finally, Sects. III D and III E briefly discuss how performing truncations applies for more general systems with many modes and beyond the long-wavelength approximation, respectively (Note that the full descriptions for these generalized systems are found in Sect. VI).