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We study four-dimensional superconformal field theories coupled tothree-dimensional superconformal boundary or defect degrees of freedom.Starting with bulk N=2, d=4 theories, we construct abelian models preservingN=2, d=3 supersymmetry and the conformal symmetries under which theboundary/defect is invariant. We write the action, including the bulk terms, inN=2, d=3 superspace. Moreover we derive Callan-Symanzik equations for thesemodels using their superconformal transformation properties and show that thebeta functions vanish to all orders in perturbation theory, such that themodels remain superconformal upon quantization. Furthermore we study a modelwith N=4 SU(N) Yang-Mills theory in the bulk coupled to a N=4, d=3hypermultiplet on a defect. This model was constructed by DeWolfe, Freedman andOoguri, and conjectured to be conformal based on its relation to an AdSconfiguration studied by Karch and Randall. We write this model in N=2, d=3superspace, which has the distinct advantage that non-renormalization theoremsbecome transparent. Using N=4, d=3 supersymmetry, we argue that the model isconformal.Comment: 30 pages, 4 figures, AMSLaTeX, revised comments on Chern-Simons term,  references adde

Four-dimensional superconformal theories with interacting boundaries or defects