Muir was born in Glasgow in 1819. His father died in 1820, when his mother moved the family to Kilmarnock. He attended Glasgow and Edinburgh Universities but before graduating his uncle secured a writership for Muir with the East India Company. He attended Haileybury before departing for India in 1837. Muir was stationed in the North West Provinces where he met and married his wife, Elizabeth, in 1840. By 1847 he was secretary to board of revenue of the North West Provinces based in Agra. In 1852 he became secretary to the Lt. Governor, James Thompson. He developed an interest in Islam Studies. He also learnt Persian and Arabic, and it was for this that he received the Gold Medal in 1903.
In 1867 he was created a Knight Commander of the Star of India, and in 1868 he became lieutenant-governor of the North-West Provinces. Muir himself founded Muir Central College in 1873. In 1887, this became the University of Allahabad. Muir served from 1874 until 1876 as financial member of the Governor-General's Council. He retired in 1876, when he became a member of the Council of India in London.
In 1885 he was elected principal of Edinburgh University. In 1884, Muir was elected President of the Royal Asiatic Society, also serving as Vice-President from 1885-1886, and 1894-1897.
His chief books are A Life of Mahomet and History of Islam to the Era of the Hegira; Annals of the Early Caliphate; The Caliphate: Its rise, decline and fall, an abridgment and continuation of the Annals, which brings the record down to the fall of the caliphate on the onset of the Mongols; The Koran: its Composition and Teaching; and The Mohammedan Controversy, a reprint of five essays published at intervals between 1885 and 1887.
Mügül Ataç Andrews, was an expert on oriental embroidery, and with her husband Peter Alford Andrews, dedicated much of her life to studying Eurasian nomadic and urban tents. She married Peter in 1967 and they had two sons.