In this paper we analyze the transport of radionuclides through penetrations in nuclear waste containers. Penetrations may result from corrosion or cracks and may occur in the original container material, in degraded or corroded material, or in deposits of corrosion products. We do not consider how these penetrations occur or the characteristics of expected penetrations in waste containers. We are concerned only with the analytical formulation and solutions of equations to predict rates of mass transfer through penetrations of specified size and geometry. Expressions for the diffusive mass transfer rates through apertures are presented. We present numerical illustrations for steady-state mass-transfer rates through a circular hole, including concentration isopleths. The results are extended to multiple holes, including a criterion for hole spacing wherein superposition of single-hole solutions can be used. Results illustrated for holes in thin-walled containers show that significant mass transfer can occur even if a small fraction of the container area is perforated. We also illustrate the case of holes facing a water gap, instead of being in intimate contact with porous rock. In this case the radionuclide flux from many small holes approaches that from a bare waste cylinder.