I. A Mathematical Physics Introduction. The Obstacle Problem. Some Free Boundary Problems. Some Mathematical Tools. II. Unilateral Elliptic Variational Inequalities. Variational Inequalities in Hilbert Spaces. Smoothness of the Variational Solution. The Coincidence Set and the Free Boundary. Unilateral Plateau Problems. III. Applications in Mechanics and in Physics. Applied Obstacle Problems. Dam and Stefan Type Problems. Bibliography. Index.