Kollman obtained his B.A. from Grinnell College in 1966 and his M.A. and Ph.D. from Princeton University in 1967 and 1970 respectively. His PhD supervisor was Leland C. Allen, who had received his PhD in 1956 from MIT supervised by John C. Slater. After a post-doctoral position at the University of Cambridge with David Buckingham, Kollman was hired as an assistant professor by UCSF, where he spent the rest of his career.
In 1995, he was distinguished with the Computers in Chemistry Award from the American Chemical Society.
He was awarded the UCSF medal in 2018.