Al-Azhar
The Cairo mosque-university founded by Shia Fatimids in 970 that became the closest thing Sunni Islam has to a central authority.

Al-Azhar is a mosque, a university, and a religious authority in Cairo, and it is the nearest thing the Sunni world has to a Vatican, with the crucial difference that its rulings bind nobody. When Western governments want to know what mainstream Sunni Islam says about something, they ask al-Azhar. When Egyptian governments want a religious argument for a policy, they ask al-Azhar. Its opinions carry weight across a quarter of humanity and no enforcement power at all, which makes it the perfect illustration of how religious authority actually works in Sunni Islam.
Founded by the wrong sect
The origin is a permanent irony. Al-Azhar was built between 970 and 972 by the Fatimids, an Ismaili Shia dynasty who had conquered Egypt and founded Cairo as the capital of a rival caliphate to Sunni Baghdad. The mosque was designed as the intellectual engine of Ismaili missionary work, training the propagandists who would carry Fatimid Shiism across the Muslim world.
It lasted two centuries in that role. Then Saladin ended the Fatimid caliphate in 1171 and converted Egypt back to Sunni Islam, and al-Azhar with it. Under the Mamluks it grew into the leading Sunni teaching institution in the world, drawing students from Morocco to Central Asia and organizing them into residential colleges by region of origin. It survived the Ottoman conquest, Napoleon's occupation, and the British.
That trajectory tells you something about Islamic institutions. They are durable in a way their sponsoring dynasties are not. The Fatimids are gone, the Ayyubids are gone, the Mamluks and the Ottomans are gone, and al-Azhar is still teaching.
What it actually does
Al-Azhar trains scholars, issues fatwas through its Islamic Research Academy and the associated office of the Grand Mufti of Egypt, certifies religious curricula, and sends thousands of imams and teachers to mosques across Africa and Asia. Since 1961 it has also been a full modern university with faculties of medicine and engineering alongside the religious sciences.
At its head is the Grand Imam, the Shaykh al-Azhar, who has no equivalent in the rest of Sunni Islam. He is not a pope. He cannot excommunicate, and no Muslim outside Egypt is obliged to follow him. What he has is prestige accumulated over a thousand years, which is a real form of power even without a mechanism.
Al-Azhar's characteristic positions matter geopolitically. It teaches the four Sunni legal schools rather than one, and it has historically recognized Twelver Shia jurisprudence as a legitimate school, a position formalized in a 1959 declaration by the Shaykh al-Azhar that Sunnis may follow Ja'fari law. That makes it the main institutional counterweight to Saudi-funded Salafi teaching, which regards Shia Islam as heresy. When Riyadh and Cairo compete for religious influence in Africa or South Asia, this is the substance of the competition.
The independence question
The recurring drama of modern al-Azhar is its relationship with the Egyptian state. Nasser nationalized it in 1961, seizing its endowments, expanding it into a state university, and making the Grand Imam a presidential appointee. The move destroyed the institution's financial independence and turned it into an arm of government, and every Egyptian president since has used it to bless official policy.
That dependence cuts both ways. Al-Azhar's endorsement is worth less because everyone knows it is not freely given, which is exactly the charge Islamist movements level at it: that the scholars have been bought. It is also, more quietly, an institution that can drag its feet, and its clerics have resisted presidential pressure on specific questions more often than the caricature allows.
Its most consequential recent role has been as the designated refuter of jihadism. Al-Azhar issues the detailed scholarly rebuttals of Islamic State theology, and the group in turn declares al-Azhar apostate. That exchange is the clearest picture of religious authority in Sunni Islam today: the oldest and most prestigious institution in the tradition arguing with an insurgency that simply denies it has standing, and no court anywhere that can rule between them.