Compact Stein surfaces with boundary as branched covers of $\B^4$
  We prove that Stein surfaces with boundary coincide up to orientation preserving diffeomorphisms with simple branched coverings of $\B^4$ whose branch set is a positive braided surface. As a consequence, we have that a smooth oriented 3-manifold is Stein fillable iff it has a positive open-book decomposition. 
Andrea Loi and Riccardo Piergallini