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Understanding the photoluminescence (PL) characteristics and exciton properties of hybrid organic-inorganic perovskites (HOIPs) plays a key role in optimizing their performance in optoelectronic devices. In particular, PL is an important technique for the investigation of carrier-phonon interactions and quantification of defects in a system. However, these properties are inaccessible when important signatures are masked by the wide inhomogeneous broadening seen in polycrystalline films, which are commonly used in devices. Here, we study PL on high quality MAPbI₃ single crystals in the orthorhombic phase. The high crystalline quality and well controlled defect density allow us to obtain PL spectra with narrow features and reveal new ones that were not resolvable before. Temperature dependence and excitation intensity dependence data clearly show features of the free exciton and several bound excitons, with consistent energy levels verified among several samples. Assignment of each peak will be discussed, as well as exciton binding energy and exciton-phonon couplings. These newly resolved fine features on high quality single crystals will allow further in-depth studies of the intrinsic photophysics and defect properties related to growth conditions in HOIPs.