The Gross-Pitaevskii (GP) equation is named after Eugene P. Gross and Lev P. Pitaevskii who introduced it independently in 1961 for the purpose of describing vortices in superfluids. At that time the only known superfluid was liquid helium, but since the first experimental realization of Bose-Einstein condensates in trapped gases of alkali atoms in 1995 the GP equation has become a basic tool for theoretical investigations of the manifold quantum phenomena exhibited by dilute, ultracold Bose gases. Mathematically, the GP equation is a special case of a non-linear Schrödinger equation and the mathematics and physics literature about it is vast. In the talk the focus will be on the following topics: 1) The interpretation of the equation on the basis of many-body quantum mechanics. 2) Vortices in rapidly rotating trapped superfluids at strong coupling and the determination of critical speeds that mark phase transitions from single vortices to a vortex lattice pattern and finally to a giant vortex.