  In this paper, we mainly review recent results on mathematical theory and
numerical methods for Bose-Einstein condensation (BEC), based on the
Gross-Pitaevskii equation (GPE). Starting from the simplest case with
one-component BEC of the weakly interacting bosons, we study the reduction of
GPE to lower dimensions, the ground states of BEC including the existence and
uniqueness as well as nonexistence results, and the dynamics of GPE including
dynamical laws, well-posedness of the Cauchy problem as well as the finite time
blow-up. To compute the ground state, the gradient flow with discrete
normalization (or imaginary time) method is reviewed and various full
discretization methods are presented and compared. To simulate the dynamics,
both finite difference methods and time splitting spectral methods are
reviewed, and their error estimates are briefly outlined. When the GPE has
symmetric properties, we show how to simplify the numerical methods. Then we
compare two widely used scalings, i.e. physical scaling (commonly used) and
semiclassical scaling, for BEC in strong repulsive interaction regime
(Thomas-Fermi regime), and discuss semiclassical limits of the GPE. Extensions
of these results for one-component BEC are then carried out for rotating BEC by
GPE with an angular momentum rotation, dipolar BEC by GPE with long range
dipole-dipole interaction, and two-component BEC by coupled GPEs. Finally, as a
perspective, we show briefly the mathematical models for spin-1 BEC, Bogoliubov
excitation and BEC at finite temperature.
