We compute analytically and numerically the four-point correlation function that characterizes nontrivial cooperative dynamics in glassy systems within several models of glasses: elastoplastic deformations, mode-coupling theory (MCT), collectively rearranging regions (CRR's), diffusing defects, and kinetically constrained models (KCM's). Some features of the four-point susceptibility chi(4) (t) are expected to be universal: at short times we expect a power-law increase in time as t(4) due to ballistic motion (t(2) if the dynamics is Brownian) followed by an elastic regime (most relevant deep in the glass phase) characterized by a t or sqrt[t] growth, depending on whether phonons are propagative or diffusive. We find in both the beta and early alpha regime that chi(4) approximately t(mu), where mu is directly related to the mechanism responsible for relaxation. This regime ends when a maximum of chi(4) is reached at a time t= t(*) of the order of the relaxation time of the system. This maximum is followed by a fast decay to zero at large times. The height of the maximum also follows a power law chi(4) (t(*)) approximately t(*lambda). The value of the exponents mu and lambda allows one to distinguish between different mechanisms. For example, freely diffusing defects in d=3 lead to mu=2 and lambda=1 , whereas the CRR scenario rather predicts either mu=1 or a logarithmic behavior depending on the nature of the nucleation events and a logarithmic behavior of chi(4) (t(*)) . MCT leads to mu=b and lambda=1/gamma , where b and gamma are the standard MCT exponents. We compare our theoretical results with numerical simulations on a Lennard-Jones and a soft-sphere system. Within the limited time scales accessible to numerical simulations, we find that the exponent mu is rather small, mu<1 , with a value in reasonable agreement with the MCT predictions, but not with the prediction of simple diffusive defect models, KCM's with noncooperative defects, and CRR's. Experimental and numerical determination of chi(4) (t) for longer time scales and lower temperatures would yield highly valuable information on the glass formation mechanism.