β-Ga₂O₃ thin films were grown on the substrates of sapphire, GaN, and single crystals of β-Ga₂O₃, using plasma-assisted molecular beam epitaxy. By varying deposition conditions, pure-phase epitaxial β-Ga₂O₃ thin films were obtained, and the crystal quality of the as-grown films was optimized. A systematic characterization and a detailed analysis were performed on the films, including the nucleation process, surface morphology, crystal quality, thermal stability, as well as electrical and optical properties. Optical absorption was investigated using photothermal deflection spectroscopy, which provides detailed information about sub-gap optical absorption. Photocurrent measurements indicated a pronounced persistent photo-conductivity of β-Ga₂O₃. A blue-UV emission with an energy of 3-3.5 eV was observed by cathodoluminescence spectroscopy. The Fermi level position of the as-grown film was determined based on temperature-dependent electrical conductivity measurements. It is proposed that oxygen vacancies in the film form a defect band at around E c-0.8 eV that pins the Fermi level and is related to the observed photocurrent and cathodoluminescence characteristics.