Very recently, two-dimensional MoSi2N4 has been synthetized (Y.-L. Hong, Z. Liu, L. Wang, T. Zhou, W. Ma, C. Xu, S. Feng, L. Chen, M.-L. Chen and D.-M. Sun, Chemical vapor deposition of layered two-dimensional MoSi2N4 materials, Science, 2020, 369, 670-674.). In this work, we systematically explore the mechanical, electronic, and catalytic properties of the MX2Y4 (M = Cr, Hf, Mo, Ti, W, Zr; X = Si, Ge; Y = N, P, As) monolayers by first-principles calculations. These observed monolayers exhibit an isotropic Young's moduli of 165-514 N m-1 and a Poisson's ratio of 0.26-0.33. The calculated band structures indicate that their bandgaps are in the range of 0.49-2.05 eV at the HSE06 level. In particular, a high electron mobility of about 1.04 × 104 cm2 V-1 s-1 is observed in TiSi2N4 monolayers, which shows potential for high-speed electronic devices. MX2Y4 monolayers also reveal decent performances in the hydrogen evolution reaction. More importantly, the Gibbs free energy change of the TiSi2N4 (ZrSi2N4) monolayer is as small as 0.078 eV (-0.035 eV), even being comparable with that of Pt (-0.09 eV). This investigation suggests that the MoSi2N4 family monolayers have potential advanced applications such as photocatalytic, electrocatalytic, and photovoltaic devices.