<p>This work is concerned with the quantitative stability of the feasible and optimal solutions of convex semi-infinite optimization problems in different parametric settings. Regarding the constraint system, we deal with convex inequalities under right-hand side (RHS), affine, and free perturbations of the original data. In these three frameworks, our focus is on quatifying the local variation of feasible solutions with respect to data perturbations (which could come, for instance, from rounding in computational processes). We formalize our analysis through the so-called Aubin property and the computation of the corresponding Lipschitz modulus. In a second stage, we tackle the Aubin property of the optimal set mapping under canonical perturbations (i.e., linear perturbations of the objective function together with RHS perturbations of the constraint system). Special attention to didactical aspects is paid; in particular, to the technics used to derive the main results gathered in this survey. Specifically, a supremum function approach and a linearization strategy based on the Fenchel-Legendre conjugate are discussed. Combining both strategies the paper provides a unified treatment (with a new proof) for the Lipschitz moduli of the feasible set mapping under RHS and affine perturbations.</p>