He was born in Boston, Massachusetts. He attended Harvard University, where he received his Bachelor's degree in 1955, his Master's degree in 1956, and his Ph.D in applied mathematics in 1959. He then worked at the IBM Thomas J. Watson Research Center. In 1968, he became Professor of Computer Science, Mathematics, and Operations Research at the University of California, Berkeley. Apart from a 4-year period as a professor at the University of Washington, he has remained at Berkeley.

His citation for the Turing Award was as follows:

- For his continuing contributions to the theory of algorithms including the development of efficient algorithms for network flow and other combinatorial optimization problems, the identification of polynomial-time computability with the intuitive notion of algorithmic efficiency, and, most notably, contributions to the theory of NP-completeness. Karp introduced the now standard methodology for proving problems to be NP-complete which has led to the identification of many theoretical and practical problems as being computationally difficult.