Approximation and interpolation employing radial basis functions has found important applications since the early 1980's in areas such as signal processing, medical imaging, as well as neural networks. Several applications demand that certain physical properties be fulfilled, such as a function being divergence free. In this talk, we will introduce a new class of matrix-valued radial basis functions that are divergence free as well as compactly supported. No such class of radial basis functions that reflects these physical properties and is of compact support had been constructed before. It leads to the possibility of applying fast solvers for inverting interpolation matrices, as these matrices are not only symmetric and positive definite, but also sparse because of this compact support. We will present error bounds and stability estimates which hold for a broad class of functions. We will conclude the talk with applications to the numerical solution of the Navier-Stokes equation for certain incompressible fluid flows.