The theory of dense graphs (and hypergraphs) has many components which are very closely connected:

1) Notion of quasi randomness and related norms,

2) local structure defined by sampling,

3) global structure defined by regularization,

4) limit theory with appropriate limit objects.

In this talk we present an analogous theory (with all the above components) for functions (or subsets) on abelian groups instead of graphs. In this theory quasi randomness is measured by the so-called Gowers norms. As a byproduct we obtain a general inverse theorem for Gowers norms which implies various results by Green Tao and Ziegler. The abelian theory is surprisingly rich algebraically and topologically. For example limit objects of functions on cyclic groups are functions on nilmanifolds which are interesting geometric objects. One of our main goals is to show that a higher order generalization of Fourier analysis is crucial to the subject. This theory deals with morphisms between certain algebraic structures first studied by Host and Kra. (Note that ordinary Fourier analysis deals with morphisms from abelian groups to the circle group U(1).)