The Cauchy-Riemann operator for domains in a complex manifold is well understood for domains in complex spaces. However, much less is known for the solvability and regularity for the Cauchy-Riemann operator in a complex manifold which is not complex spaces  or Stein. Recently, some progress has been made for the L2 theory of the Cauchy-Riemann equations on product domains in complex manifolds. An analogous formula of the classical Kunneth formula for the harmonic forms are also obtained. We have also discuss an L2 version of the Serre duality for domains on complex manifolds. Furthermore, duality between the harmonic spaces and the Bergman space in complex manifolds will also be presented (Joint work with Debraj Chakrabarti).