  In this paper, we develop a general existence theory for properly embedded
minimal surfaces with free boundary in any compact Riemannian 3-manifold $M$
with boundary $\partial M$. The main feature of our result is that no convexity
assumption is required on $\partial M$. Our proof uses a variant of the min-max
construction first considered by Almgren and Pitts. Recently, Colding-De Lellis
gave a simplified proof of the interior regularity and here, we prove the
boundary regularity of the limiting embedded minimal surfaces at their free
boundaries. In addition, we define a topological invariant, the filling genus,
for compact 3-manifolds with boundary and show that we can bound the genus of
the minimal surface constructed above in terms of the filling genus of the
ambient manifold $M$.
