[1] He has been noted for contributions to C*-algebras, quantum field theory, partial differential equations, fluid dynamics, scientific computing, and the modeling of petroleum reservoirs.

Together with Arthur Jaffe, he has founded a subject called constructive quantum field theory.

[3] More recently, the United States Department of Energy adopted Glimm's front-track methodology for shock-wave calculations, e.g., simulating weapons performance.

[6] In 1993, Glimm was awarded the Leroy P. Steele Prize for his contribution to solving hyperbolic systems of partial differential equations.

[8] Starting January 1, 2007, he served a 2-year term as president of the American Mathematical Society.