where N is the total number of molecules coupled to the cavity mode, and the collective normalized coupling strength η characterizes the light-matter coupling strength. Setting N = 1 will go back to the singlemolecule case. Note that the above relation between Ω R and η only holds under the linear approximation of the dipole operator, and it breaks down for ultra-strong coupling (USC) regime and beyond when η > 0.1. While the scaling of the Rabi-splitting with the number of molecules N is well understood theoretically and verified experimentally, it is not clear how VSC modification of chemical reactivity could depend on N . Currently, there is no cohesive theory that fully explains the range of phenomena experimentally observed for VSC reactions in the collective regime. However, many groups have made important and notable advances to this field that hopefully further elucidate the problem at hand and inspire future advances in the field. With that in mind, the rest of this section discusses many of these creative theoretical advances in VSC, reviewing the methods, results, and drawbacks of each one of these theories.