The concept of an Asymptotic-Preserving (AP) method makes a breakthrough in the numerical resolution of multiscale differential problems. AP schemes are extremely powerful tools as they permit the use of the same scheme to discretize a multiscale problem and its asymptotic limit reduced problem, with fixed discretization parameters. This allows to match regions where the multiscale parameter has very different orders of magnitude. In this case, the AP scheme realizes an automatic transition between the multiscale problem and its limit problem. We survey some recent results concerning the development of such schemes for various multiscale problems, like hyperbolic systems with stiff relaxation terms and kinetic equations in fluid regimes.