Levi-flat hypersurfaces arise naturally from complex foliation theory and complex dynamics. In this talk we will discuss the Cauchy-Riemann equations on domains in complex manifolds with Levi-flat boundary. We give an example of a pseudoconvex domain in a complex manifold whose $L^2$-Dolbeault cohomology is non-Hausdorff, yet the domain is Stein with real-analytic Levi-flat boundary (Joint work with Debraj Chakrabarti).