Ellis was the youngest of six children of Francis Ellis (1772–1842) of Bath and his wife Mary. Educated privately, he entered Trinity College, Cambridge, in 1836, graduating as Senior Wrangler in 1840 and elected Fellow of Trinity shortly afterwards.^[1] Although he had also entered the Inner Temple in 1838, was called to the bar in 1840, and later helped William Whewell with jurisprudence, Ellis never practised law. He hoped unsuccessfully for the Cambridge chair of civil law.

Inheriting substantial Irish estates when his father died, Ellis contemplated entering Parliament as a Whig under Sir William Napier's patronage. Yet his courtship of one of Napier's daughters ended in some confusion: Ellis never married, and never stood for Parliament.

As a mathematician, Ellis founded the Cambridge Mathematical Journal with D. F. Gregory in 1837. He corresponded with Augustus De Morgan on the conjectured four color theorem.

Continental travel failed to restore Ellis' health. An attack of rheumatic fever at Sanremo in 1849 left him an invalid, and he returned to Cambridge, living at Anstey Hall, Trumpington, next to his friend John Grote, vicar of Trumpington. From his sickbed Ellis kept up contact with the young Trinity mathematician William Walton, and dictated his thoughts on a wide range of topics, including etymology, bees' cells, Roman money, the principles of a projected Chinese dictionary, and Boole's The Laws of Thought (1854).

Ellis took on the editing of Francis Bacon's works with two other Trinity fellows, Douglas Denon Heath and James Spedding. Dramatic deterioration of Ellis's health from 1847 left his work on the general prefaces to Bacon's philosophy unfinished. Spedding and Heath completed the Works in seven volumes, published 1857–1859.

Ellis's own major mathematical contributions were on functional and differential equations, and the theory of probability ("On the foundations of the theory of probabilitiesW, read to the Cambridge Philosophical Society on 14 February 1842; published in the fourth volume of the Proceedings of the Cambridge Philosophical Society of 1844). Philosophically, Ellis, like George Boole and later John Venn, defended an objective rather than subjective theory of probability.

William Walton edited a posthumous collection of both published and unpublished writings, in The mathematical and other writings of R. L. Ellis (1863): this was prefaced by a biographical memoir by Harvey Goodwin. Correspondence and notebooks of Ellis are amongst the Mayor Papers and Whewell Papers at Trinity College, Cambridge.^[2]

Leslie translated Dante, Roman law texts and Danish ballads; a gentle melancholia suffuses the lines of his own poetry which he left in manuscript.

• Goodwin, Harvey (1863) "Biographical Memoir of Robert Leslie Ellis". In W. Walton, ed., The Mathematical and other Writings of R. L. Ellis.
• Kiliç, Berna (2000) "Robert Leslie Ellis and John Stuart Mill on the one and the many of frequentism", British Journal for the History of Philosophy 8:2.
• Panteki, Maria, "Ellis, Robert Leslie (1817–1859)", Oxford Dictionary of National Biography, Oxford University Press, 2004.
• Verburgt, Lukas M. (2013) "Robert Leslie Ellis's work on philosophy of science and the foundations of probability theory", Historia Mathematica 40:4.
• Verburgt, Lukas M. (2022) [1] A Prodigy of Universal Genius: Robert Leslie Ellis, 1817–1859. Springer.

• Works by Robert Leslie Ellis at Project Gutenberg
• Works by or about Robert Leslie Ellis at Internet Archive
• Works by Robert Leslie Ellis at LibriVox (public domain audiobooks)  
• The Mathematical and Other Writings of Robert Leslie Ellis Edited by William Walton. With a biographical memoir by the Very Reverend Harvey Goodwin. (London: Deighton, Bell & Co., Bell and Daldy, 1863.)
• "Robert Leslie Ellis" entry at the MacTutor History of Mathematics archive