Peter Kronheimer has made outstanding contributions to a number of areas, ranging from complex geometry to manifold topology. Much of his early work centred on ‘hyperkahler’ structures in differential geometry, which are special solutions of the Einstein field equations. Extending the work of Nigel Hitchin, Peter constructed and classified all asymptotically locally Euclidean hyperkähler manifolds, uncovering a beautiful and intricate theory which combines group theory, algebraic and differential geometry. Soon after, he discovered a family of hyperkähler structures on complex coadjoint orbits. Since 1990, Peter’s work has been focused on 4-manifold topology. In a long collaboration with Tomasz Mrowka, he developed the theory of singular solutions of the Yang–Mills equations and applied it to obtain far-reaching results on 4-manifold invariants, and particularly to solve long-standing problems about surfaces in 4-manifolds. This work paved the way for the introduction of the Seiberg–Witten invariants, a development to which Peter has made fundamental contributions.