Single quantum emitters (SQEs) are at the heart of quantum optics(1) and photonic quantum-information technologies(2). To date, all the demonstrated solid-state single-photon sources are confined to one-dimensional (1D; ref. 3) or 3D materials(4-7). 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 similar to 130 mu eV, about two orders of magnitude smaller than those of delocalized valley excitons(8). Second-order correlation measurements revealed a strong photon antibunching, which unambiguously established the single-photon nature of the emission(9). The SQE emission shows two non-degenerate transitions, which are cross-linearly polarized. We assign this fine structure to two excitonic eigen-modes whose degeneracy is lifted by a large similar to 0.71 meV coupling, probably because of the electron-hole exchange interaction in the presence of anisotropy(10). Magneto-optical measurements also reveal an exciton g factor of similar to 8.7, several times larger than those of delocalized valley excitons(11-14). 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.