He was born in Budapest, but has spent his working life in the USA. He was Professor at Harvard University 1959-1999, and received the Wolf Prize in 2000,
Initially he worked on the theory of electrical circuits (Bott-Duffin theorem from 1949), then switched to pure mathematics.
He studied the homotopy theory of Lie groups, using methods from Morse theory, leading to the Bott periodicity theorem (1956).
This led to his role as collaborator over many years with Michael Atiyah, initially via the part played by periodicity in K-theory; he made important contributions towards the Index Theorem, especially in formulating related fixed-point theorems (in particular the so-called 'Woods Hole' fixed-point theorem).
He is also known in connection with the Borel-Bott-Weil theorem on representation theory of Lie groups via holomorphic sheaves and their cohomology groups; and for work on foliations.