We propose product kernels for the approximation of high-dimensional functions from scattered data. Product kernels are tensors of component kernels, each acting on a low-dimensional Euclidean space, whereby they provide efficient and flexible tools for multivariate scattered data approximation. In this talk, we first discuss theoretical properties of product kernels, before we address relevant computational aspects. To this end, we explain how the structural properties of product kernels can be exploited to obtain adaptive and data-driven kernel methods for high-dimensional approximation. Supporting numerical examples show the good performance of the proposed kernel approximation method. This talk is based on joint work with Kristof Albrecht and Juliane Entzian.