Woodward was born in Boston, Massachusetts, the son of Arthur Woodward (an immigrant from England) and Margaret Woodward, nee Burns (an immigrant from Scotland, born in Glasgow).
At an early age, Woodward was attracted to chemistry and engaged in private study while he attended the public primary and secondary schools of Quincy, Massachusetts. In 1933, he entered the Massachusetts Institute of Technology (MIT), but neglected his formal studies badly enough to be expelled the next year. MIT readmitted him in 1935, and by 1936 he had received the Bachelor of Science degree. Only one year later, MIT awarded him the doctorate. He took a postdoctoral fellowship at neighboring Harvard University from 1937 to 1938, and remained at Harvard in various capacities for the rest of his life.
While still remaining at Harvard, he took on the directorship of the Woodward Research Institute, based at Basel, Switzerland in 1963. He also became a trustee of his alma mater, MIT, from 1966 to 1971 and of the Weizmann Institute of Science in Israel.
In 1965 he received the Nobel Prize in Chemistry for his synthetic work with natural products (including such molecules as chlorophyll and many antibiotics). That same year, he and his student Roald Hoffmann devised rules (now called the Woodward-Hoffmann rules) for elucidating the stereochemistry of the products of organic reactions. Woodward formulated his ideas (which were based on the symmetry properties of molecular orbitals) based on his experiences as a synthetic organic chemist; he asked Hoffman to perform theoretical calculations to verify these ideas, which were done using Hoffmann's Extended Hückel method. Hoffmann received the 1981 Nobel Prize for this work; Woodward undoubtedly would have received a second Nobel Prize as well, except that Nobel Prizes are not awarded posthumously.
He continued working on organic syntheses, developing a process for the synthesis of vitamin B-12 in 1971.
Woodward received numerous awards from scientific societies and honorary degrees, and was made a member or honorary member of a large number of academies all over the world.
In 1938 he married Irja Pullman, and in 1946 he married Eudoxia Muller. From the first marriage he had two daughters, and from the second one daughter and one son.
He died in Cambridge, Massachusetts, while working on the synthesis of an antibiotic, erythromycin.