Integral geometry started from the observation that the theory of representations of $SL(2,\Bbb C)$ is a consequence of the fact that the family of horocycles of this group can be considered as the family of lines in $\Bbb C^3$ intersecting hyperbola. Since hyperbola in this construction can be replaced by any other curve (where are no groups at all), the basic challenge of integral geometry is to discover geometrical structures( more general than group ones) where it is possible to develop harmonic analysis.
Most complete picture is known for integral geometry of complex curves. It turns out that Radon type inversion formulas exist for complete families of rational curves and only for them. It is remarkable that such families universally appear in explicitly solvable problems of nonlinear analysis ( twistors). In integral geometry there are known also some results where curves replace by submanifolds of bigger dimension. It corresponds to some integrable nonlinear problems with several spectral parameters. It would be interesting to understand them in this more broad context.