Roberts was elected a Member of the American Academy of Arts and Sciences in 1952. He was elected Member of the National Academy of Sciences in 1956 at 38 years old. In 1978, he was elected a Fellow of The Explorers Club. In 1984, Roberts received the Golden Plate Award of the American Academy of Achievement, awarded the Priestley Medal in 1987, the National Medal of Science in 1990, the Glenn T. Seaborg Medal in 1991, the NAS Award in Chemical Sciences in 1999, the Nakanishi Prize in 2001, the NAS Award for Chemistry in Service to Society in 2009, the Linus Pauling Legacy Award in 2006 and the American Institute of Chemists Gold Medal in 2013.