Rapoport was born in Lozоvaya, Russia. In 1922, he came to the United States, and in 1928 he became a naturalized citizen. He started studying music in Chicago and continued with piano, conducting and composition at the Vienna Hochschule für Musik, where he studied from 1929 to 1934. However, due to the rise of Nazism, he found it impossible to make a career as a pianist.

He shifted his career into mathematics, getting a Ph.D. degree in mathematics under Nicolas Rashevsky at the University of Chicago in 1941. According to the Toronto Globe and Mail, he was a member of the American Communist Party for three years, but quit before enlisting in the U.S. Army Air Corps in 1941, serving in Alaska and India during World War II.

After the war, he joined the Committee on Mathematical Biology at the University of Chicago (1947 – 1954), publishing his first book, Science and the Goals of Man, co-authored with semanticist S.I. Hayakawa in 1950. He also received a one year fellowship at the prestigious Center for Advanced Study in the Behavioral Sciences (Stanford, California).

From 1955 to 1970 Rapoport was Professor of Mathematical Biology and Senior Research Mathematician at the University of Michigan, as well as founding member, in 1955, of the Mental Health Research Institute (MHRI) at the University of Michigan. In 1970 Rapoport moved to Toronto to avoid the war-making ways of the Vietnam-era United States. He was appointed professor of mathematics and psychology at the University of Toronto, 1970 - 1979. He lived in bucolic Wychwood Park overlooking downtown Toronto, a neighbour of Marshall McLuhan. On his retirement from the University of Toronto, he became director of the Institute of Advanced Studies (Vienna) until 1983.

In 1954, Anatol Rapoport cofounded the Society for General Systems Research, along with the researchers Ludwig von Bertalanffy, Ralph Gerard, and Kenneth Boulding. He became president of the Society for General Systems Research in 1965.

Anatol Rapoport died of pneumonia in Toronto. He is survived by his wife Gwen, daughter Anya, and sons Alexander and Anthony.

Rapoport contributed to general systems theory, mathematical biology and to the mathematical modeling of social interaction and stochastic models of contagion. He combined his mathematical expertise with psychological insights into the study of game theory, social networks and semantics. Rapoport extended these understandings into studies of psychological conflict, dealing with nuclear disarmament and international politics. His autobiography, Certainties and Doubts: A Philosophy of Life, was published in 2001.

Rapoport had a versatile mind, working in mathematics, psychology, biology, game theory, social network analysis, and peace and conflict studies. For example, he pioneered in the modeling of parasitism and symbiosis, researching cybernetic theory. This went on to give a conceptual basis for his lifelong work in conflict and cooperation.

Among many other well-known books on fights, games, violence and peace, Rapoport was the author of over 300 articles and of Two-Person Game Theory (1966) and N-Person Game Theory (2001). He analyzed contests in which there are more than two sets of conflicting interests, such as war, diplomacy, poker or bargaining. His work led him to peace research, including books on The Origins of Violence (1989) and Peace, An Idea Whose Time Has Come (1993), both written at the University of Toronto.

He won a computer tournament in the 1980s, based on Robert Axelrod's The Evolution of Cooperation. This sought to understand how cooperation could emerge through evolution. Rapoport's entry, Tit-For-Tat has only four lines of code. The program opens by cooperating with its opponent. It then plays exactly as the other side played in the previous game. If the other side defected in the previous game, the program also defects; but only for one game. If the other side cooperates, the program continues to cooperate. According to Peace Magazine author/editor Metta Spencer, the program "punished the other player for selfish behaviour and rewarded her for cooperative behaviour — but the punishment lasted only as long as the selfish behaviour lasted. This proved to be an exceptionally effective sanction, quickly showing the other side the advantages of cooperating. It also set moral philosophers to proposing this as a workable principle to use in real life interactions."

Anatol Rapoport was an early developer of social network analysis. His original work showed that one can measure large networks by profiling traces of flows through them. This enables learning about the speed of the distribution of resources, including information, and what speeds or impedes these flows — such as race, gender, socioeconomic status, proximity and kinship. This work linked social networks to the diffusion of innovation, and by extension, to epidemiology. Rapoport's empirical work traced the spread of information within a school. It prefigured the study of Six degrees of separation, by showing the rapid spread of information in a population to almost all — but not all — school members.

According to Thomas Homer-Dixon in the Toronto Globe and Mail, Rapoport "became anti-militarist quite soon after the war. The idea of military values became anathema." He was a leading organizer of the first teach-ins against the Vietnam War at the University of Michigan, a model that spread rapidly throughout North America. He told at a teach-in: "By undertaking the war against Vietnam, the United States has undertaken a war against humanity.... This war we shall not win." (Ann Arbor News, April 1967). He said he was an abolitionist, rather than a total pacifist: "I'm for killing the institution of war".

Rapoport returned to the University of Toronto to become the founding (and unpaid) Professor of Peace and Conflict Studies programme, working with George Ignatieff and Canada's Science for Peace organization. As its sole professor at the start, he used a rigorous, interdisciplinary approach to the study of peace, integrating mathematics, politics, psychology, philosophy, science and sociology. His main concern was to legitimize peace studies as a worthy academic pursuit. The Trudeau Centre for Peace and Conflict Studies continued to flourish at the University of Toronto under the leadership of Thomas Homer-Dixon, and, from 2008, under Ron Levi. When Rapoport began, there were one (unpaid) professor and twelve students. Now, there are three (paid) professors and ninety students.

Rapoport's students report that he was an engaged and inspiring professor who captured their attention, imagination and interest with his wide-ranging knowledge, passion for the subject, good humor, kind and generous spirit, attentiveness to student concerns and animated teaching style.

In 1981, Rapoport co-founded the international NGO Science for Peace, and in 1984 he created the famous tit for tat strategy for the iterated prisoner's dilemma tournament held by Robert Axelrod that year. He was recognized in the 1980s for his contribution to world peace through nuclear conflict restraint via his game theoretic models of psychological conflict resolution. He won the Lenz International Peace Research Prize in 1976.