Borwein has made substantial contributions to distinct mathematical disciplines: optimization theory and practice, number theory, classical analysis, theory of computation, and functional analysis. The fundamental Borwein-Preiss smooth variational principle in optimization theory enables classical techniques to be applied to highly nonsmooth problems. Borwein's cubic analogue of Jacobi's seminal 19th century quartic theta function identity is celebrated in number theory. Borwein is also a world leader in the field of experimental mathematics using intensive computation for pure mathematical discovery. Highly successful as supervisor, expositor, and populariser of mathematics, Borwein authored with E.J. Borowski the best selling modern dictionary of mathematics.