Single quantum emitters (SQEs) are at the heart of quantum optics and photonic quantum-information technologies. To date, all the demonstrated solid-state single-photon sources are confined to one-dimensional (1D; ref. 3) or 3D materials. Here, we report a new class of SQEs based on excitons that are spatially localized by defects in 2D tungsten-diselenide (WSe₂) monolayers. The optical emission from these SQEs shows narrow linewidths of ∼130μeV, about two orders of magnitude smaller than those of delocalized valley excitons. Second-order correlation measurements revealed a strong photon antibunching, which unambiguously established the single-photon nature of the emission. The SQE emission shows two non-degenerate transitions, which are cross-linearly polarized. We assign this fine structure to two excitonic eigenmodes whose degeneracy is lifted by a large ∼0.71meV coupling, probably because of the electron-hole exchange interaction in the presence of anisotropy. Magneto-optical measurements also reveal an exciton g factor of ∼8.7, several times larger than those of delocalized valley excitons. In addition to their fundamental importance, establishing new SQEs in 2D quantum materials could give rise to practical advantages in quantum-information processing, such as an efficient photon extraction and a high integratability and scalability.