  This thesis aims to explore the structure of CFTs with global internal
symmetries and beyond via the Large-Charge Expansion (LCE), a semi-classical
expansion applicable for states with large global quantum numbers. In the first
part of this thesis we study CFT and SSB. We discuss the symmetry-constraints
imposed by conformal invariance, introduce the concept of CFT data and the OPE.
Concerning SSB, we discuss the existence of NG modes, the general counting rule
for the number of NG modes under the spontaneous breaking of global internal
symmetries and a generalization of the Goldstone theorem at finite density. In
the second part of this thesis we discuss the current state-of-the-art
understanding of the LCE and systematically study CFTs with a global O(2)
symmetry in the context of the LCE. We discuss the emergence of effective
condensed-matter descriptions, in particular superfluids, in correlators
involving states with large global quantum numbers. We then use the superfluid
EFT description to systematically study two-, three- and four-point functions
for CFTs with a global O(2) symmetry. In the last part of this thesis we study
CFTs in the double-scaling limit of large charge and large N . We discuss the D
= 3 WF fixed point at large N and derive the leading order asymptotics at large
charge Q in the double scaling limit Q/N fixed. Next, we use a fixed-charge
approach to gain access to the leading order effective potential for the
quartic theory, which we then study for spacetime dimensions 2 < D < 6. In D =
3, we reproduce and extend old results originally found by re-summing Feynman
diagrams. Finally, we discuss the interacting fixed points of three-dimensional
fermionic CFTs in the double-scaling limit. While the GN model exhibits a
Fermi-sphere description at large charge, whose fate at finite N is yet to be
determined, for the NJL-type models we find a BEC.
