Abstract: The problem of relating the geometric structure of an underlying measure μ with the L²(μ)-boundedness of singular integrals, where the archetypal example is the action of the Cauchy transform on 1-dimensional measures, was initiated by Calderón in 1977. Since then the topic has been studied widely and had deep impact in harmonic, complex and potential analysis, leading to the resolution of long standing open problems such as Vitushkin's conjecture, which was proved by David. While the Cauchy transform is quite well understood not many things are known for other kernels. We will outline some open problems in the area, describe the reasons behind this gap in our knowledge and discuss several recent advances involving different kernels.