This bachelor thesis deals with the Stefan problem, from its historical background to the existence and uniqueness of solution to the problem. The physical background is presented at the beginning. Afterwards there are presented some results related to the problem, like explicit solutions and an analysis of the technique for obtaining solutions to the problem by perturbation methods. Also the theoretical development and mathematical formulation of supercooled Stefan problems is included in this part. Finally, results on existence and uniqueness for the Stefan problems are shown. In particular, the cases treated here are the ones concerning small and large times, both for Dirichlet boundary conditions, and Neumann boundary conditions for small times.