Abhyankar was born in a Maharashtrian Konkanastha Brahmin family. He earned his B.Sc. from Royal Institute of Science in 1951, his A.M. at Harvard University in 1952, and his Ph.D. at Harvard in 1955. His thesis, written under the direction of Oscar Zariski, was titled Local uniformization on algebraic surfaces over modular ground fields. Before going to Purdue, he was an associate professor of mathematics at Cornell University and Johns Hopkins University.
Shreeram was appointed the Marshall Distinguished Professor of Mathematics at Purdue in 1967. His research topics include algebraic geometry (particularly resolution of singularities, a field in which he made significative progress over fields of finite characteristic), commutative algebra, local algebra, valuation theory, theory of functions of several complex variables, quantum electrodynamics, circuit theory, invariant theory, combinatorics, computer-aided design, and robotics. He popularized the Jacobian conjecture.