John Butcher’s research has centred on numerical methods for solving differential equations. In particular he has contributed to the theory of Runge-Kutta methods and to the formulation and theory of general linear methods. Based on the combinatorial approach to the order conditions for Runge-Kutta methods, he proposed the use of what has become known as the Butcher Group, for the study of compositions of properties of various numerical approximations. It is interesting that a related structure, now known to be a Hopf Algebra, has modern applications ranging from non-commutative geometry to renormalisation of Feynman integrals in theoretical Physics. An equivalent formulation of John’s algebraic approach, by other authors and known as B-series, has become of central importance in the study of numerical methods for a variety of evolutionary problems. General linear methods, after many years, have become of practical importance by the discovery of new sub-families of numerical schemes which seem to have advantage over traditional numerical algorithms.