 Casson has worked in both high-dimensional manifold topology and 3- and 4-dimensional topology,
using both geometric and algebraic techniques.Among other discoveries, he contributed 
to the disproof of the manifold Hauptvermutung, introduced the Casson invariant, a modern invariant for 3-manifolds, and Casson handles, used in Michael Freedman's proof of the 4-dimensional Poincaré conjecture.In 1991, he was awarded the Oswald Veblen Prize in Geometry by the American Mathematical Society.