Lyons received his PhD in 1970 at the University of Chicago under John Griggs Thompson with a thesis entitled Characterizations of Some Finite Simple Groups with Small 2-Rank. From 1972 to 2017, he was a professor at Rutgers University.
With Daniel Gorenstein and Ronald Solomon he wrote, and is continuing to write, now with Inna Capdeboscq, a series on the second-generation proof of the classification program for finite simple groups. Nine volumes of this series have been published so far. He discovered a sporadic group which Charles Sims constructed and called the Lyons group Ly.
In 2012, he shared the Leroy P. Steele Prize for Mathematical Exposition, awarded by the American Mathematical Society, with Michael Aschbacher, Stephen D. Smith, and Ronald Solomon. In 2013, he became a fellow of the American Mathematical Society "for contributions to the classification of the finite simple groups, including the discovery of one of the 26 sporadic finite simple groups.".
- with Gorenstein: The local structure of finite groups of characteristic 2 type, American Mathematical Society, 1983
- with Daniel Gorenstein, Ronald Solomon: The classification of the finite simple groups, American Mathematical Society, 9 Vols., 1994–2021
- with Michael Aschbacher, Stephen D. Smith, Ronald Solomon: The classification of finite simple groups: Groups of characteristic 2 type, American Mathematical Society, 2011
- "Mathematicians who classified finite simple groups". Archived from the original on September 24, 2011. Retrieved March 28, 2012.
- Richard Lyons at the Mathematics Genealogy Project
- Richard Lyons (1972,5) "Evidence for a new finite simple group", Journal of Algebra 20:540–569, and 34:188–189
- List of Fellows of the American Mathematical Society