We extend to all orders in perturbation theory a method to calculate supersymmetry-breaking effects by analytic continuation of the renormalization group into superspace. A central observation is that the renormalized gauge coupling can be extended to a real vector superfield, thereby including soft breaking effects in the gauge sector. We explain the relation between this vector superfield coupling and the "holomorphic" gauge coupling, which is a chiral superfield running only at one loop. We consider these issues for a number of regulators, including dimensional reduction. With this method, the renormalization group equations for soft-supersymmetry-breaking terms are directly related to supersymmetric beta functions and anomalous dimensions to all orders in perturbation theory. However, the real power of the formalism lies in computing finite soft breaking effects corresponding to high-loop component calculations. We prove that the gaugino mass in gauge-mediated supersymmetry breaking is "screened" from strong interactions in the messenger sector. We present a complete next-to-leading calculation of gaugino masses (two loops) and sfermion masses (three loops! in minimal gauge mediation and several other calculations of phenomenological relevance. [S0556-2821(98)06319-X].