  In a recent work, new two-dimensional materials, the monolayer
MoSi$_{2}$N$_{4}$ and WSi$_{2}$N$_{4}$, have been successfully synthesized in
experiment, and several other monolayer materials with the similar structure,
such as MoSi$_{2}$As$_{4}$, have been predicted [{\color{blue}Science 369,
670-674 (2020)}]. Here, based on first-principles calculations and theoretical
analysis, we investigate the electronic and optical properties of monolayer
MoSi$_{2}$N$_{4}$, WSi$_{2}$N$_{4}$ and MoSi$_{2}$As$_{4}$. We show that these
materials are semiconductors, with a pair of Dirac-type valleys located at the
corners of the hexagonal Brillouin zone. Due to the broken inversion symmetry
and the effect of spin-orbit coupling, the valley fermions manifest spin-valley
coupling, valley-contrasting Berry curvature, and valley-selective optical
circular dichroism. We also construct the low-energy effective model for the
valleys, calculate the spin Hall conductivity and the permittivity, and
investigate the strain effect on the band structure. Our result reveals
interesting valley physics in monolayer MoSi$_{2}$N$_{4}$, WSi$_{2}$N$_{4}$ and
MoSi$_{2}$As$_{4}$, suggesting their great potential for valleytronics and
spintronics applications.
