In this work, a complete moment convergence theorem is obtained for weighted sums of asymptotically almost negatively associated (AANA) random variables without assumption of identical distribution under some mild moment conditions. As an application, the complete convergence theorems for weighted sums of negatively associated (NA) and AANA random variables are obtained. The result not only generalizes the corresponding ones of Sung \cite{15} and Huang et al. \cite{16}, but also improves them.