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Rafiyev was born in Ganja, Azerbaijan in 1888. Son of local philanthropist Haji Mammadhuseyn Rafiyev, he received his elementary education at the madrasah of Shah Abbas Mosque and then at Ganja Classic Gymnasium. Musa Rafiyev had 6 brothers and 5 sisters. In 1911, he graduated from the Medical Department of Kharkov State University and returned to Ganja to start his medical practice. For exceptional services, he received several awards, among them "Exemplar Council" (1913), "College Assessor" (1914), "Saray Council" (1916). In 1914, in cooperation with Khudadat Rafibeyli and Hasan bey Aghayev, he co-founded the first Medical Society of Elisabethpol.