The Radon-Carleman Problem (in honor of the pioneering work of J. Radon and T. Carleman to singular integral operators in potential theory) is a broad label for questions having to do with computing or estimating the essential norm and the Fredholm radius of singular integral operators of double layer type, associated with elliptic operators on function spaces naturally intervening in the formulation of boundary value problems for said partial differential operators. In this talk I will review the genesis of this topic and present a series of results in increasingly more irregular settings, culminating with that of uniformly rectifiable domains. The main goal is to monitor how the geometry of the domain affects the complexity of this type of study. This is joint work with Dorina Mitrea and Marius Mitrea from Baylor University.