We review some mathematical results on the Ginzburg–Landau model with and without magnetic ﬁeld. The Ginzburg–Landau energy is the standard model for superconductivity, able to predict the existence of vortices (which are quantized, topological defects) in certain regimes of the applied magnetic ﬁeld. We focus particularly on deriving limiting (or reduced) energies for the Ginzburg–Landau energy functional, depending on the various parameter regimes, in the spirit of Γ-convergence. These passages to the limit allow to perform a sort of dimension-reduction and to deduce a rather complete characterization of the behavior of vortices for energy-minimizers, in agreement with the physics results. We also describe the behavior of energy critical points, the stability of the solutions, the motion of vortices for solutions of the gradient-ﬂow of the Ginzburg–Landau energy, and show how they are also governed by those of the limiting energies.