In this work we present a family of methods designed to represent implicitly- defined hypersurfaces satisfying prescribed constraints. These methods arise rather naturally from a theoretical framework of generalized interpolation in function spaces induced by certain radial basis functions (RBFs). We begin by presenting some results from the theory of optimal recovery in Hilbert spaces and Hermite-Birkhoff interpolation using RBFs. After the mathematical pre- liminaries, we introduce Hermite Radial Basis Functions (HRBF) Implicits as a representation for implicit surfaces appearing naturally from the special case of (first-order) Hermite interpolation with RBFs. HRBF Implicits reconstruct an implicit function which interpolates or approximates scattered multivariate Hermite data (i.e. unstructured points and their corresponding normals) and its theory unifies a recently introduced class of surface reconstruction methods based on RBFs which incorporate normals directly in their problem formulation. This class has the advantage of not depending on manufactured offset-points to ensure existence of a non-trivial RBF interpolant. This framework not only allows us to show connections between the present method and others but also enables us to enhance the flexibility of this method by ensuring well-posedness of an interesting combined interpolation/regularisation approach. Experiments suggest that HRBF Implicits allow the reconstruction of surfaces rich in details and behave better than previous related methods under coarse and/or nonuni- form samplings, even in the presence of close sheets. Following our presentation of HRBF Implicits, we present other formulations which relax the assumptions on the nature of the datasets. For instance, we begin by relaxing a coherence re- quirement on the input normals and present two different approaches to recover implicitly-defined hypersufaces from points and normal-directions, one which only solves a linear system and another based on an eigenvalue problem. After that, we show a formulation which does not require normals but still recon- structs a nontrivial implicit function by computing the “optimal” normals in a rather natural sense through the solution of another eigenvalue problem. The last formulation assumes the dataset consists of on-surface points and corre- sponding tangent-directions, a natural scenario in surface reconstruction from contours and sketch-based modelling systems. We conclude with results from a number of experiments and a discussion on future perspectives and further investigations.