Quantum chaotic systems are often defined via the assertion that their spectral statistics coincides with, or is well approximated by, random matrix theory. In this talk I will explain how the universal content of random matrix theory emerges as the consequence of a simple symmetry-breaking principle and its associated Goldstone modes. This approach gives a natural way to identify wormhole-like correlations, even for individual theories, in particular in theories with gravity duals. I will also discuss how to extend the Goldstone effective-field-theory approach to study operator correlation functions, and explain the relation of the EFT of quantum chaos to the bulk physics of wormhole-like geometries. [Please email alejandro.cabo_bizet@kcl.ac.uk for the Zoom link]