Random matrix theory (RMT) universality is the defining property of quantum mechanical chaotic systems, and can be probed by observables like the spectral form factor (SFF). In this talk, I’ll describe systematic deviations from RMT behaviour at intermediate time scales in systems with approximate symmetries. In such systems, the approximate symmetries allow us to organize the Hilbert space into approximately decoupled sectors. At early times, each of which contributes independently to the SFF. At late times, the SFF transitions into the final ramp of the fully mixed chaotic Hamiltonian. The transitional behaviour is governed by a universal process that we call Hilbert space diffusion, with a diffusion constant related to the relaxation rate of the associated nearly conserved charge.