  We study the asymptotics of almost holomorphic sections $s \in H^0_J(M,
\omega)$ of an ample line bundle $L \to M$ over an almost complex symplectic
manifold in the sense of Boutet de Monvel-Guillemin. Such sections are defined
as the kernel of a complex which is analogous to the $\bar{\partial}$ complex
for a positive line bundle over a complex manifold. Our main result is the
scaling limit asymptotics of the Szego projectors $\Pi_N$ of powers $L^N$. The
Kodaira embedding theorem and Tian almost isometry theorem are almost immediate
consequences of the scaling limit. We also relate such almost holomorphic
sections to the asymptotically holomorphic sections in the sense of Donaldson
and Auroux.
