"Much of the structure of the world around us is currently understood in the language of quantum field theory. As such theories have been highly tested by experiment we have a great deal of confidence that they are correct, at least in their regimes of applicability. However, to answer many of the most important open questions, from the nature of confinement to the origin of high temperature superconductivity, we need to develop new tools to find exact, non-perturbative results. For specific gauge theories we can understand their strongly coupled behaviour by making use of their dual description in terms of certain quantum theories of gravity – the so-called gauge/gravity duality.The paradigmatic example is the conjectured duality between four dimensional maximally supersymmetric Yang-Mills and strings on a curved space-time, five dimensional anti-de Sitter space. Recently, the discovery that this example of the gauge/gravity duality is integrable, i.e. that there are an infinite number of hidden conserved quantities, in the planar limit has led to a great deal of progress, for example providing exact results for the anomalous dimensions of all operators.In the proposed project the origin and structure of this integrability is to be considered with an eye to developing tools to calculate further interesting observables at all values of the coupling. In particular, extending the techniques of integrable models to the calculation of three-point functions would, in principle, give a complete determination of the planar theory. It is also proposed to study a wider class of observables and, for example, the exact form of planar scattering amplitudes, which are related to higher point correlation functions, is an open problem whose solution may well be within our theoretical grasp. Furthermore, the methods and results will be generalised towards other, more phenomenologically relevant, theories."