Financial Mathematics has been one of the major growth areas of mathematics over the last few decades. It involves the mathematical modelling of economic agents and their interactions and incorporates mathematical tools from a wide range of mathematical disciplines including functional analysis, numerical analysis and pdes, stochastic analysis and control, and statistics. Brownian motion, and more generally probability theory, provide the language for describing the random characteristics we observe in economic behaviour. `A misplaced reliance on sophisticated mathematics' was recently cited, perhaps provocatively, as a contributory cause of the credit crunch. Nonetheless, the finance industry, with the UK as a major international centre, continues to demand innovative mathematics to support the trading of derivative securities on the interest rate, foreign exchange and credit markets. Sophisticated mathematics is now embedded in finance, from both the academic and practitioner viewpoints, so the key goals are the development of new more reliable models, an improved understanding of the mathematics that underpin them, and an appreciation of their limitations.