  Let S be a closed connected real surface and f a smooth embedding or
immersion of S into a complex surface X. Assuming that the number of complex
points of the immersion (counted with algebraic multiplicities) is non-positive
we prove that f can be uniformly approximated by an isotopic immersion g whose
image g(S) in X has a basis of open Stein neighborhoods which are homotopy
equivalent to g(S). We obtain precise results for surfaces in the complex
projective plane CP^2 and find an immersed symplectic sphere in CP^2 with a
Stein neighborhood. Conversely, the generalized adjunction inequality for
embedded oriented real surfaces in complex surfaces shows that the existence of
a Stein neighborhood implies non-positivity of the number of complex points.
