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Gallium oxide (b-Ga2O3) is an emerging semiconductor with relevant properties for power electronics, solar-blind photodetectors, and some sensor applications due to its ultra-wide bandgap and developing technology base for high quality, melt-based substrate growth and thick, low-doped homoepitaxial layers. Of critical importance for the commercialization of this potentially important material is understanding of doping mechanisms in the monoclinic lattice, where two types of Ga sites and three types of O sites have been identified. A critical literature review of doping and defects of the monoclinic b-phase of gallium oxide is provided in this work. Theoretical fundamentals of both donor and acceptor doping in Ga2O3 are reviewed. Advances in doping of epitaxial Ga2O3 with a focus on molecular beam epitaxy and ion implantation are critically examined. As doping is fundamentally related to defects, particularly in this material, a review of defect characterization by optical and electrical spectroscopic methods is provided as well. P-type doping, one of the fundamental challenges for Ga2O3, is discussed in terms of first-principles calculations and ion implantation of known acceptors such as Mg and N.