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New well-conditioned frequency and time domain integral equations providing rapidly convergent solutions to electromagnetic radiation and scattering problems have been obtained. By leveraging (i) novel integral identities, (ii) frequency and time domain Calderon formulas, and (iii) novel wavelet bases, the proposed constructs permit the resonant-free and accurate analysis of ultrabroadband and multiscale electromagnetic phenomena that hitherto resisted analysis by all known simulation methodologies. The new formulations have been applied successfully to the analysis of real-life problems, including the characterization of integrated circuit interconnects, the design of broadband antennas, and the simulation of electromagnetic interactions with spacecraft; effciency gains ranging between twenty and fifty with respect to previously available solvers, were obtained. This, together with the high degree of integrability of the presented techniques into the existing technology, implies they could rapidly impact the state of the art in boundary element solvers in use in academia and industry. The ideas presented here have applications that go beyond the scope of this thesis; future studies will analyze their extension to volume/surface integral equations, to high order and singular basis functions, and to their hybridization with finite element technology.