Born at Nicolson Street in Edinburgh on 5 November 1773, Monro received his M.D. from Edinburgh in 1797, then studied in London under Wilson and in Paris, returning to Edinburgh in 1800.
He was elected a Fellow of the Royal Society of Edinburgh in 1798, his proposers being Andrew Duncan, John Hill and Thomas Charles Hope.
In the early 19th century Edinburgh University was regarded as the best medical school in the United Kingdom but had declined significantly from its heyday in the Enlightenment of the 18th century. Two thirds of the professors were appointed by the Tory-controlled Edinburgh Corporation on the basis of their party list subject to approval by the Kirk, with little regard for ability. In some cases families treated the university chairs as hereditary, and critics alleged that Alexander Monro III exemplified the "mediocrity" this could produce. His manner was described as "unimpassioned indifference" and lectures were known to degenerate into riots.
Monro took little pride in his personal appearance and was described by contemporaries as dishevelled, scruffy and even dirty. This was an era when many in medicine considered cleanliness to be finicking and affected. "An executioner might as well manicure his nails before chopping off a head." For this reason, Charles Darwin, a student at Edinburgh University in 1825, was disgusted by Monro arriving at lectures still bloody from the dissecting room. Darwin wrote his family that "I dislike [Monro] and his lectures so much that I cannot speak with decency about them. He is so dirty in person and actions." Many students turned to competing private schools in Surgeon's Square instead, with Charles' brother Erasmus going to John Lizars, but Charles found the sight of surgery so upsetting that he stopped trying and turned his attention to natural history.
During Monro's tenure as Professor of Anatomy, Edinburgh was rocked by scandal due to the notorious "Burke and Hare murders" in which healthy individuals were intentionally killed in order to supply cadavers for dissection by anatomy lecturers and their students. One of the murderers, William Burke, was hanged on 28 January 1829, after which he was famously dissected at the Edinburgh Medical College by Monro himself. In a letter, Monro dipped his quill pen into Burke's blood and wrote, "This is written with the blood of Wm Burke, who was hanged at Edinburgh. This blood was taken from his head."
Alexander Monro Tertius resigned as the Chair of Anatomy in 1846 and thus ended the dynastic reign of Monros at Edinburgh University which had spanned 126 years. Among Monro's publications are "Outlines of the Anatomy of the Human Body" (1811) in four volumes and "Elements of Anatomy" (1825) in two volumes. Although he taught surgery but had never trained or practised as a surgeon. He was Secretary of the Royal College of Physicians of Edinburgh from 1809 to 1819 and elected President in 1825 and 1826. He was also on the Council of Wernerian Natural History Society of which he became a member in 1811.
In 1841 Dr Robert Halliday Gunning came to Edinburgh to oversee Monro's anatomy rooms and work as his assistant.
Monro died at Craiglockhart, south-west of Edinburgh on 10 March 1859 and is buried in Lord's Row against the western wall of Dean Cemetery.
He is known as "tertius" because his two predecessors as professor of anatomy at Edinburgh University had the same name: these were his grandfather (known as Alexander Monro primus) and his father (known as Alexander Monro secundus). Alexander's great-grandfather, John Munro, was also in the medical profession.
He married twice: firstly, in 1800, to Maria Agnes Carmichael-Smyth (1776-1833), the daughter of Dr. Carmichael-Smyth, by whom he had twelve children; and secondly, in 1836, to the daughter of David Hunter. The latter survived him.
In the 1830s he was living, with his large family and first wife, at 1 Great Stuart Street on the Moray Estate in Edinburgh's west end. The house stands on a prominent corner partly facing the gardens of Moray Place. Monro's neighbour (at 3 Great Stuart Street) was Dr Robert Christison.
His son Sir David Monro made a career as a politician in New Zealand, and was the second Speaker of the New Zealand House of Representatives.
In the 2010 motion picture Burke and Hare, Monro is bitter rivals with Robert Knox (Tom Wilkinson) whom he thwarts at every turn by having a statute passed ensuring all dead bodies be passed on to him for dissection. He also has an unhealthy obsession with feet. Monro is portrayed by Tim Curry.