We will discuss a collection of results centered around the mapping properties of the X-ray transform $I_0$ and its normal operator on simple Riemannian manifolds with boundary. Such results depend on choices of weighted $L^2$ topologies, to be chosen on the manifold of interest and its space of geodesics. In the cases presented, appropriate scales of Sobolev spaces can be derived, where to formulate accurate mapping properties for $I_0$ and $I_0^* I_0$, in particular allowing for sharp stability estimates. Some more new interesting facts will be discussed along the way in certain geometries: new connections between $I_0^* I_0$ and degenerate elliptic differential operators; anisotropic Sobolev scales on the space of geodesics which capture the smoothing properties of the X-ray transform, and applications to exact filtered-backprojection formulas.