Quasi-K?hler manifolds with trivial Chern Holonomy

In this paper we study almost complex manifolds admitting a quasi-K?hler Chern-flat metric (Chern-flat means that the holonomy of the Chern connection is trivial). We prove that in the compact case such manifolds are all nilmanifolds. Some partial classification results are established and we prove that a quasi-K?hler Chern-flat structure can be tamed by a symplectic form if and only if the ambient space is isomorphic to a flat torus.