  We present the asymptotic transitions from microscopic to macroscopic
physics, their computational challenges and the Asymptotic-Preserving (AP)
strategies to efficiently compute multiscale physical problems. Specifically,
we will first study the asymptotic transition from quantum to classical
mechanics, from classical mechanics to kinetic theory, and then from kinetic
theory to hydrodynamics. We then review some representative AP schemes that
mimic, at the discrete level, these asymptotic transitions, hence can be used
crossing scales and, in particular, capture the macroscopic behavior without
resolving numerically the microscopic physical scale.
