Altman attended the Massachusetts Institute of Technology (B.S., 1960, in physics) and the University of Colorado (Ph.D., 1967, in biophysics). He was a molecular biology fellow at Harvard University (1967–69) and at the University of Cambridge (1969–70) before joining the biology faculty at Yale University in 1971. He became a full professor at Yale in 1980, was the department chairman from 1983 to 1985, and served as dean of the undergraduate Yale College from 1985 to 1989. He took U.S. citizenship in 1984 but concurrently retained his Canadian citizenship.

Working independently, Altman and Cech discovered a new role for RNA. The old belief was that enzymatic activity—the triggering and acceleration of vital chemical reactions within living cells—was the exclusive domain of protein molecules. Altman's and Cech's revolutionary discovery was that RNA, traditionally thought to be simply a passive carrier of genetic codes between different parts of the living cell, could also take on active enzymatic functions. This new knowledge opened up new fields of scientific research and biotechnology and caused scientists to rethink old theories of how cells function.