Smalley's research in physical chemistry investigated the formation of inorganic and semiconductor clusters using pulsed molecular beams and time-of-flight mass spectrometry. As a consequence of this expertise, Robert Curl introduced him to Harry Kroto in order to investigate a question about the constituents of astronomical dust.  These are carbon-rich grains expelled by old stars such as R Coronae Borealis. The result of this collaboration was the discovery of C (known as Buckyballs) and the fullerenes as the third allotropic form of carbon.
Smalley recognized that the structure of C was like that of a soccer ball after cutting and tapping hexagons together in a three-dimensional manner, utilizing 20 hexagons and 12 pentagons. He was also responsible for the name of C, naming it after Buckminster Fuller, an American architect who was known for his use of geodesic domes in his designs. 
The research that earned Kroto, Smalley and Curl the Nobel Prize mostly comprised three articles.  First was the discovery of C in the November 14, 1985, issue of Nature, "C: Buckminsterfullerene". The second article detailed the discovery of the endohedral fullerenes in "Lanthanum Complexes of Spheroidal Carbon Shells" in the Journal of the American Chemical Society (1985).  The third announced the discovery of the fullerenes in "Reactivity of Large Carbon Clusters: Spheroidal Carbon Shells and Their Possible Relevance to the Formation and Morphology of Soot" in the Journal of Physical Chemistry (1986).
Although only three people can be cited for a Nobel Prize, graduate students James R. Heath, Yuan Liu, and Sean C. OBrien participated in the work. Smalley mentioned Heath and OBrien in his Nobel Lecture. Heath went on to become a professor at the California Institute of Technology (Caltech) and O'Brien joined Texas Instruments and is now at MEMtronics.  Yuan Liu is a Senior Staff Scientist at Oak Ridge National Laboratory.
This research is significant for the discovery of a new allotrope of carbon known as a fullerene.  Other allotropes of carbon include graphite, diamond and graphene. Harry Krotos 1985 paper entitled "C60: Buckminsterfullerine", published with colleagues J. R. Heath, S. C. OBrien, R. F. Curl, and R. E. Smalley, was honored by a Citation for Chemical Breakthrough Award from the Division of History of Chemistry of the American Chemical Society, presented to Rice University in 2015. The discovery of fullerenes was recognized in 2010 by the designation of a National Historic Chemical Landmark by the American Chemical Society at the Richard E. Smalley Institute for Nanoscale Science and Technology at Rice University in Houston, Texas.