About 50 years ago I.Gelfand suggested that the horospherical
transform is an universal tool to solve problems of harmonic
analysis on homogeneous spaces. Gelfand and Graev gave several spectacular illustrations of this methods but simultaneously several serious restrictions of the method were clear. The most important of them is that discrete series of representations liein the kernel of the horospherical transform and can not be investigated in such a way.
I want to show that the union of the horospherical method with complex analysis essentially extends the possibilities of the method. Firstly, I want to explain how complex horospheres work for compact groups Lie (where there are now real horospheres) and I believe that it gives an interesting view on this classical subject. I will also talk about several more advanced applications: the separation of series of representations on pseudo Riemannian symmetric spaces, harmonic analysis on nonsymmetric homogeneous spaces, connections with non linear differential equations.