In the recent couple of years, we have achieved an incredibly deep understanding of N=4 maximally supersymmetric gauge theory and the AdS/CFT correspondence which relates this model to string theory. The main reason for this progress consists in the apparent exact integrability of the models in the planar limit. Integrability is a hidden symmetry which allows to establish very efficient tools for performing calculations. Remarkably, these tools not only conveniently compute observables at very high orders in the coupling constant, both at weak and at strong coupling, but they also make quantitative predictions at finite coupling strength. A similar amount of progress is due to the development of novel on-shell techniques which allow to construct scattering amplitudes at several loop orders. They become especially powerful when combined with the extended symmetries related to integrability.The aim of the project is to put the recent rapid progress in integrability and scattering amplitudes on a solid foundation. By enhancing the encountered symmetries and applications towards more realistic gauge and gravity theories we hope to obtain new tools for QFT in general as well as new clues for the problem of quantum gravity.More concretely, we will work out a precise formulation for the algebra underlying integrability. This is a crucial step towards proving integrability in AdS/CFT and to justify and develop efficient methods. Furthermore, we plan to develop applications of integrability away from the planar limit and for non-integrable gauge theories. Finally, we will extend these methods and considerations to gravity models. We will also take a fresh look at alternative models with a view to solving the puzzle of quantum gravity.We plan to address these important objectives with the common framework of extended symmetries and powerful calculational techniques for scattering amplitudes.