Integral geometry is a field of mathematics that studies inversions and various properties of transforms, which integrate functions along curves, surfaces and hypersurfaces. Such transforms arise naturally in numerous problems of medical imaging, remote sensing, and non-destructive testing. The most typical examples include the Radon transform and its generalizations. The talk will discuss some problems and recent results related to generalized Radon transforms, and their applications to various problems of tomography. Behind many of the recent advances in number theory, like Fermat's last theorem or the Sato-Tate conjecture, are Galois representations and their associated L-functions. Galois representations are of great interest to number theorists since they encode the answers to many arithmetic questions.