The 2D form of the BeO sheet has been successfully prepared (Hui Zhang et al., ACS Nano, 2021, 15, 2497). Motivated by these exciting experimental results on the 2D layered BeO structure, we studied the effect of the adsorption of B atoms on BeO (B@BeO) and substitutional B atoms (B-BeO) at the Be site at different B concentrations. We investigated the structural stability and the mechanical, electronic, magnetic, and optical properties of the mentioned structures using first-principles calculations. We found out that hexagonal BeO monolayers with adsorbed and dopant B atoms have different mechanical stabilities at different concentrations. B@BeO and B-BeO monolayers are brittle structures, and B@BeO structures are more rigid than B-BeO monolayers (at the same B concentration). The adsorption and the formation energy per B atom decrease as the B concentration increases. In comparison, the work function increases when increasing the B concentration. The work function of B@BeO is higher than the corresponding value of B-BeO (at the same B concentration). The magnetic moment linearly increases as the B concentration increases. BeO is a semiconductor with an indirect bandgap of 5.3 eV. The B@BeO and B-BeO structures are semiconductors, except for 3B-BeO (14.2% doped concentration), which is a metal. The bandgap is 1.25 eV for most of the adsorbed atom concentrations. For B-BeO, the bandgap decreases to zero at a concentration of 14.2%. The bandgap of the B-BeO monolayer at different B concentrations is smaller than the corresponding values of the B@BeO monolayer, which indicates that B substitutional doping has a greater effect on the electronic structure of the BeO monolayer than B adsorption doping. We investigated the optical properties, including the dielectric function and absorption coefficient. The results indicate good optical absorption in the range of infrared and ultraviolet energies for the B adsorbed and doped BeO monolayer.