The broad range of applications of copper, including areas such as electronics, fuel cells, and spent nuclear fuel disposal, require accurate description of the physical and chemical properties of copper compounds. Within some of these applications, cuprous hydroxide is a compound whose relevance has been recently discovered. Its existence in the solid-state form was recently reported. Experimental determination of its physical-chemical properties is challenging due to its instability and poop crystallinity. Within the framework of density functional theory calculations (DFT), we investigated the nature of bonding, electronic spectra, and optical properties of the cuprous oxide and cuprous hydroxide. It is found that the hybrid functional PBE0 can accurately describe the electronic structure and optical properties of these two copper(I) compounds. The calculated properties of cuprous oxide are in good agreement with the experimental data and other theoretical results. The structure of cuprous hydroxide can be deduced from that of cuprous oxide by substituting half Cu+ in Cu2O lattice with protons. Compared to Cu2O, the presence of hydrogen in CuOH has little effect on the ionic nature of Cu–O bonding, but lowers the energy levels of the occupied states. Thus, CuOH is calculated to have a wider indirect band gap of 2.73 eV compared with the Cu2O band gap of 2.17 eV.