Spin glasses are statistical mechanics models where the interaction between spin dynamical variable is mediated by random couplings. The Gibbs probability measure is thus random and one is interested in studying the quenched equilibrium state. The order parameter introduced in the solution of the mean-field (Sherrington-Kirkpatrick) instance of the model is the spin overlap probability distribution for two copies of the system. A non trivial overlap distribution signals the existence of a complex equilibrium state. In this talk, after a brief review of the most recent mathematical advances in the field, we will present rigorous results for some joint probability distribution of more than 2 copies of the system which hold beyond the mean-field setting. We will also discuss numerical results for the overlap distribution of 3 copies, which is expected to be ultrametric.