In this article, we establish complete convergence theorems for weighted sums in sub-linear expectations space. As corollaries, we obtain Marcinkiewicz strong laws for weighted sums under the sub-linear expectations. Our results extend and improve the corresponding results of Bai and Cheng (Statist. Probab. Lett. 46: 105-112, 2000) and Cuzick (J. Theoret. Probab. 8: 625-641, 1995) from classical probability space to sub-linear expectation space.