This talk is an overview presentation made by D.M. Titterington as a summary of the activities at Cambridge during the Spring of 2008. Most of twentieth-century statistical theory was restricted to problems in which the number p of 'unknowns', such as parameters, is much less than n, the number of experimental units. However, the practical environment has changed dramatically over the last twenty years or so, with the spectacular evolution of computing facilities and the emergence of applications in which the number of experimental units is comparatively small but the underlying dimension is massive, leading to the desire to fit complex models for which the effective p is very large. Areas of application include image analysis, microarray analysis, finance, document classification, astronomy and atmospheric science. Some methodological advances have been made, but there is a need to provide firm consolidation in the form of a systematic and critical assessment of the new approaches as well as appropriate theoretical underpinning in this 'large p, small n' context. The existence of key applications strongly motivates the programme, but the fundamental aim is to promote core theoretical and methodological research. Both frequentist and Bayesian paradigms will be featured. The programme is directed at a broad research community, including both mainstream statisticians and the growing population of researchers in machine learning.