  In this paper we obtain the following results: (1) Any compact Stein surface
with boundary embeds naturally into a symplectic Lefschetz fibration over the
2-sphere. (2) There exists a minimal elliptic fibration over the 2-disk, which
is not Stein. (3) The circle bundle over a genus n>1 surface with euler number
e=-1 admits at least n+1 mutually non-homeomorphic simply-connected Stein
fillings. (4) Any surface bundle over the circle, whose fiber is a closed
surface of genus n>0 can be embedded into a closed symplectic 4-manifold,
splitting the symplectic 4-manifold into two pieces both of which have positive
b_2^+. (5) Every closed, oriented connected 3-manifold has a weakly
symplectically fillable double cover, branched along a 2-component link.
