This interview covers Ralph Gomory's family and upbringing, undergraduate education at Williams College, and graduate education in mathematics at Princeton University under Solomon Lefschetz. There is discussion of Gomory's time serving in the Navy, working at the Office of Naval Research, and exposure to operations research. The interview then turns to Gomory's return to Princeton, and his work on integer programming, and subsequent hiring by IBM. The interview covers Gomory's time as part of the Mathematics Department with Herman Goldstine and Benoit Mandelbrot, and under Director of Research Emanuel Piore, and also Gomory's own tenure as Director of Research from 1970 to 1989. There is some discussion of the staff he oversaw in this position, and some challenges faced by IBM Research in this period, as well as about his public writing on subjects such as the Japanese challenge to American economic productivity.