  We present an analysis of the Ginzburg-Landau equations for the description
of a two-dimensional superconductor in a bounded domain. Using the properties
of a special integrability point of these equations which allows vortex
solutions, we obtain a closed expression for the energy of the superconductor.
  The role of the boundary of the system is to provide a selection mechanism
for the number of vortices.
  A geometrical interpretation of these results is presented and they are
applied to the analysis of the magnetization recently measured on small
superconducting disks. Problems related to the interaction and nucleation of
vortices are discussed.
