This thesis presents several new techniques for rapidly converging boundary element solutions of electromagnetic problems. A special focus has been given to formulations that are relevant for electromagnetic solutions in biological tissues both at low and high frequencies. More specifically, as pertains the low-frequency regime, this thesis presents new schemes for preconditioning and accelerating the Forward Problem in Electroencephalography (EEG). The regularization strategy leveraged on a new Calderon formula, obtained in this thesis work, while the acceleration leveraged on an Adaptive-Cross-Approximation paradigm. As pertains the higher frequency regime, with electromagnetic dosimetry applications in mind, the attention of this work focused on the study and regularization of the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) integral equation via hierarchical techniques. In this effort, a complete analysis of the equation for both simply and non-simply connected geometries has been obtained. This allowed to design a new hierarchical basis regularization strategy to obtain an equation for penetrable media which is stable in a wide spectrum of frequencies. A final part of this thesis work presents a propaedeutic discretization framework and associated computational library for 2D Calderon research which will enable our future investigations in tomographic imaging.