We consider time-dependent (linear and nonlinear) Schr¨odinger equations in
a semiclassical scaling. These equations form a canonical class of (nonlinear)
dispersive models whose solutions exhibit high-frequency oscillations. The
design of efficient numerical methods which produce an accurate approximation
of the solutions, or at least of the associated physical observables, is a
formidable mathematical challenge. In this article we shall review the basic
analytical methods for dealing with such equations, including WKB asymptotics,
Wigner measure techniques and Gaussian beams. Moreover, we shall
give an overview of the current state of the art of numerical methods (most of
which are based on the described analytical techniques) for the Schr¨odinger
equation in the semiclassical regime
