The development of high-efficiency anode materials with large capacity, high stability and fast diffusion rates is a key requirement for rechargeable Li-ion and Na-ion batteries (LIBs/NIBs). In this work, the adsorption and diffusion of Li and Na atoms on two-dimensional (2D) Si3N materials is studied using first-principles calculations. The Si3N monolayers have large adsorption energies (2.74 eV for Li and 2.17 eV for Na) and a high theoretical capacity (1772.0 mAh/g for Li and 859.6 mAh/g for Na). Moreover, the low diffusion barriers (0.45 and 0.24 eV) for Li and Na atoms indicate that Si3N has an excellent high charge/discharge capability. In addition, molecular dynamics simulations showed that the structure of the 2D Si3N monolayer with adsorption of 32 Li/Na atoms has a very small change at 400 K due to the large adsorption energies for Li/Na. Owing to its good features, the Si3N monolayer is a highly promising anode material for energy storage devices.