Richard Borcherds has conducted seminal work in applying the ideas of conformal quantum field theory to solve classical problems in pure mathematics. At the same time, his work has led to a deeper understanding of conformal field theory. After important work on the classification of lattices, Richard introduced the notions of a vertex algebra, which played a key role in the construction of the natural representation of the monster group (the largest sporadic finite simple group) and a generalized Kac–Moody algebra, which led him to a proof of the moonshine conjectures of John Conway and Simon Norton. The character formula he proved for such algebras, together with the observation that their denominator functions are often automorphic forms on 0(n,2) led him to a remarkable product formula for the modular function j, amongst other results. Recently, his introduction of quantum rings, generalising vertex algebras, has provided a natural setting for the elusive concept of fusion in conformal field theory.