In this joint work with Pierre Dolbeault and Giuseppe Tomassini we consider the problem of characterizing compact real submanifolds of $C^n$ that bound Levi flat hypersurfaces. The problem is well studied in $C^2$ but surprisingly little is known in higher dimension. In this talk I will, in particular, explain the fundamental difference between $n=2$ and higher dimension showing why the known methods in $C^2$ do not apply.