Muon conversion — the process of a bound muon decaying into an energetic electron — provides one of the most stringent limits on Charged Lepton Flavor Violation. The experimental limit is soon expected to improve by four orders of magnitude, thus calling for precise theoretical predictions. While QED corrections significantly affect the shape of the spectra, their calculation is extremely challenging below the nuclear scale. By exploring modern EFT techniques (involving NRQED and SCET), I present a framework that systematically computes those corrections in a consistent and systematically improvable way. I also show that the framework is not restricted to muon conversion, but applies to a multiplicity of bound-muon decays. As a key application, I provide the most accurate predictions to date for the signal and background spectra in muon conversion.