  In 1960s, Almgren initiated a program to find minimal hypersurfaces in
compact manifolds using min-max method. This program was largely advanced by
Pitts and Schoen-Simon in 1980s when the manifold has no boundary. In this
paper, we finish this program for general compact manifold with nonempty
boundary. As a result, we prove the existence of a smooth embedded minimal
hypersurface with free boundary in any compact smooth Euclidean domain. An
application of our general existence result combined with the work of Marques
and Neves shows that for any compact Riemannian manifolds with nonnegative
Ricci curvature and convex boundary, there exist infinitely many embedded
minimal hypersurfaces with free boundary which are properly embedded.
