The Radon transform of the characteristic function of a domain in \(\mathbb R^n\) (the section function) evaluates the volumes of plane sections and is one of the basic metric characteristics in geometric tomography. We are interested in the following question: how is the geometry of a domain linked with the algebraic properties of the section function? The question is motivated by the Arnold problem (in turn, rooted in Newton’s Lemma About Ovals) on algebraically integrable domains and has been the subject of extensive study over the past decade. A brief overview of work related to the above question and recent new results on bodies with algebraic Radon transforms are presented.