This thesis is devoted to the study of the 3d Ginzburg-Landau energy of a superconducting sample occupying a thin neighborhood of a two dimensional manifold subjected to an external magnetic field. Features of minimizers of this energy are investigated, such as the appearance and location of vortices as the intensity of the applied field is increased above a certain critical value. The analysis is carried out through a dimension reduction. In the first part of this thesis it is established that the Ginzburg-Landau energy functional converges in a suitable variational sense to a functional posed on the manifold. The framework used is that of Γ-convergence. Once this is achieved, we identify the intensity of the applied field that forces the emergence of vortices in minimizers, the so-called first critical field H_c1 in Ginzburg-Landau theory, for closed simply connected manifolds and arbitrary fields. In the case of a surface of revolution and constant vertical field, we further determine the exact number of vortices and their asymptotic location in the sample as the intensity of the applied field is raised just above H_{c 1}: Finally, we derive via Γ-convergence similar statements for 3d domains of small thickness, where in this setting point vortices are replaced by vortex lines. The vortex lines are shown to consist of two collections that concentrate near the poles. Finally, it is proved that the configurations of the limiting vortices in the manifold tend to minimize a renormalized energy.