An ab initio evolutionary search algorithm was combined with density functional theory (DFT) calculations to predict a series of 2-D BxNy (1 < x/y ≤ 2). Particularly, B5N3 and B7N5 monolayers have sufficiently low formation enthalpy and excellent dynamic stability that make them promising for synthesis in experiments. Electronic structure calculations reveal that B5N3 and B7N5 monolayers possess an indirect band gap of 1.99 eV and a direct band gap of 2.40 eV, respectively. The calculated absorption coefficients for B5N3 and B7N5 monolayers are significantly improved in the low end of the visible region compared with that of 2-D h-BN. Moreover, our calculations reveal that both B5N3 and B7N5 monolayers have high electron carrier mobilities. The narrow band gaps, high carrier mobilities, strong near-ultraviolet absorption, and high synthesis possibility of B5N3 and B7N5 monolayers render them promising new materials for application in novel electronics and environmentally benign solar energy conversion.