Motivated by applications across a wide range of scientific and engineering disciplines, there has been growing interest in recent years in approximation methods for manifold-valued functions, that is, functions whose codomain has the structure of a smooth Riemannian manifold. Such maps arise, for example, in parametric model order reduction, where the codomain is typically the Stiefel or Grassmann manifold; in interpolation of diffusion tensor images obtained from magnetic resonance imaging, where the codomain is the manifold of symmetric positive definite matrices; in Cosserat-type material models, where values lie in the special Euclidean group; or in crystallographic texture analysis, where the codomain can be interpreted as a quotient of the special orthogonal group by a finite symmetry group. Radial basis function (RBF) interpolation is a well-established technique for reconstructing functions from scattered data. While the interpolation of scalar- or vector-valued functions defined on manifolds using RBFs is well understood, the RBF interpolation of functions taking values in manifolds has received comparatively little attention. In this talk, we combine RBF interpolation with three standard approaches to manifold-valued approximation: embedding-based interpolation with projection, interpolation in tangent spaces, and interpolation via Riemannian means. We derive rigorous error estimates that are analogous to those of the classical Euclidean RBF theory. The theoretical results are illustrated by numerical experiments, including both academic test cases and examples drawn from the aforementioned application areas.