The aim of the present proposal is to establish a research team developing and exploiting dualities arising in super-symmetric gauge theories, string theory and conformal field theories. These will also have many applications outside these fields. The overarching aims of the team will be: To develop established dualities into computational tools for physical quantities such as the S-matrix, correlation functions and partition functions. The construction of explicit examples of new dualities. To use such dualities to gain new insights into the mathematical structure of the theories involved.The proposal brings together researchers with different areas of expertise: super-symmetric gauge theories, string theories, conformal field theories, integrable systems and special functions. We divide it into two strands:Strand I. Deals with the AdS/CFT correspondence, scattering amplitudes and correlation functions. The main objectives are to compute scattering amplitudes of planar maximally-super symmetric Yang-Mills to all values of the coupling; extend these computations to the non-planar case; compute efficiently correlation functions in this theory.Strand II. Deals with new and exciting correspondences between four dimensional super-symmetric theories and two dimensional conformal field theories. We aim to find more examples of 4d/2d correspondences and to develop the established ones (and new ones) into efficient computational tools which will be used, for instance, to compute correlation functions in 2d Conformal Toda theories and other CFT's and even physical quantities in theories that do not admit a Lagrangian description. Progress in the first part of this strand will be used to understand the elusive 6d (2,0) theory. Furthermore, we will actively look for common mathematical structures between strands I and II.