Single quantum emitters (SQEs) are at the heart of quantum optics and photonic quantum information technologies. To date, all demonstrated solid-state single-photon sources are confined in three-dimensional materials. In this project, we found new class of SQEs based on excitons that are spatially localized by defects in two-dimensional tungsten-diselenide monolayers. The optical emission from these SQEs shows narrow linewidths of ~130 μeV, about two orders of magnitude smaller than that of delocalized valley excitons. Second-order correlation measurements reveal strong photon antibunching, unambiguously establishing 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 eigen-modes whose degeneracy is lifted by a large ~0.71 meV coupling, likely due to 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 that 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 efficient photon extraction and high integratability and scalability.