We study generalized complex (GC) manifolds from the point of view of symplectic and
Poisson geometry. We start by recalling that every GC manifold admits a canonical
Poisson structure. We use this fact, together with Weinstein's classical result on
the local normal form of Poisson, to prove a local structure theorem for GC, complex
manifolds, which extends the result Gualtieri has obtained in the "regular'' case.
Finally, we begin a study of the local structure of a GC manifold in a neighborhood
of a point where the associated Poisson tensor vanishes. In particular, we show that
in such a neighborhood, a "first-order approximation'' to the GC structure is
encoded in the data of a constant B-field and a complex Lie algebra.