The problem of determining a function from known integrals of that function over hyper-surfaces (curves in 2D case) leads to the notion of the generalized Radon transform. We treat this problem and describe the inversion procedure. It is shown that the problem of inversion of the generalized Radon transform can be reduced to solving a Fredholm integral equation. Also, we consider some applications. In particular, we estimate the error of Shepp-Logan's filter used in Computerized X-ray Tomography, present inversion formulae for Exponential and Attenuated Radon transforms, and consider the example of the Hyperbolic Radon transform.