Professor Barber is internationally known for his substantial contributions to the mathematical development of statistical mechanics. With M. E. Fisher he developed the scaling theory of systems of finite size, together with the related theory of remnant functions. His articles and reviews are widely consulted, being the leading exposition of these subjects. He pioneered the application of optimal control theory to the renormalisation group, revealing the limitations of Kadanoff' s variational approach. His renormalisation group technique for analysing Monte Carlo data is widely used. He has also made significant contributions to two-dimensional lattice models, and to improved techniques for extrapolating sequences.