Curl was educated at Rice University, Texas, and the University of California, Berkeley, where he gained his PhD in 1957. After working at Harvard he returned to Rice in 1967 as professor of chemistry.

Curl's initial work was on small clusters of atoms of semiconductors, such as germanium and silicon. In 1984, under the influence of Harry Kroto (1939–2016), he became interested in the possibility of producing long-chain carbon molecules, and persuaded his colleague Richard Smalley to deploy the resources of his laboratory towards this end. Although they expected on theoretical grounds to discover linear chain clusters with up to 33 carbon atoms, they in fact came across an unexpected molecule with 60 carbon atoms and with a cage-like structure. The discovery of this new allotrope of carbon, later named buckminsterfullerene, opened up a new branch of materials science.

Curl shared the 1996 Nobel Prize for chemistry with Smalley and Kroto.