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A Fatwā (فتوى; plural fatāwā فتاوى) is an Islamic religious ruling, a scholarly opinion on a matter of Islamic law, issued by a recognized religious authority in Islam. The Islamic jurist issuing the fatwa is called a mufti. Due to the absence of a formal religious hierarchy in Islam, adherents of the religion and Islamic governments effectively choose to follow whichever rulings they find most convincing (or, perhaps, most attractive).
Binding vs. non-binding fatwas
The dichotomy between state and scholar was absent during Muhammad's life and the early caliphates of Muhammad's companions, and the dichotomy remains absent in modern day Iran. This is because Muhammad was considered infallible, the companions who ruled as caliph after Muhammad were considered scholars of the law capable of giving (at the time informal) fatwas, and Shi‘ites hold their Imams and Ayatollahs (modern-day rulers of Iran) to be infallible as well. In these cases, the governmental rulings were binding (and while belief in agreement with the rulings of the companions was not required, compliance was - this is distinct from most others who would come to claim the title of caliph, as these subsequent caliphs throughout history would generally not be considered competent jurists or muftis capable of delivering fatwas).
Fatwa vs. court judgement (qada)
As Sunni caliphs subsequent to Muhammad's companions were generally not themselves considered muftis capable of delivering rulings on Islamic law, the Islamic court system evolved to distinguish the roles of the jurist and the judge. Jurists were generally private scholars who issued specific fatwas that carried only the weight of the issuing jurist's reputation, whereas judges, or Qadis, were state employees who delivered a binding judgement for individual cases, often referencing the fatwas of reputable jurists in the process.