The Ginzburg-Landau model is a widely used tool for describing the physical state of superconductors, superfluids, or Bose-Einstein condensates. It is also a rich source of interesting results in the calculus of variations and in the study of singularities in solutions to partial differential equations. In this talk, I will start with the basics of Ginzburg-Landau vortices, with the classical results of Bethuel-Brezis-Hélein. Then I will present some 3D results on vortex lines for the full Ginzburg-Landau model in the context of Gamma-convergence. Finally, I will present an overview of some recent results on the effect of anisotropy in the mathematical study of superconductors. Anisotropy is very important in the understanding of high temperature superconductors, and it presents very nice unexpected mathematical results.