I will introduce a parabolic flow of almost K\"ahler structures,
providing an approach to constructing canonical geometric structures on symplectic manifolds. I will exhibit this flow as one of a family of parabolic flows of almost Hermitian structures, generalizing my previous work on parabolic flows of Hermitian metrics. I will exhibit a long time existence obstruction for solutions to this flow by showing certain smoothing estimates for the curvature and torsion. Finally I will discuss the limiting objects as well as some open problems related to the symplectic
curvature flow.