Abstract: We introduce a meshless finite element framework for solving light transport problems. Traditional finite element methods use basis functions parameterized directly on the mesh surface. The creation of suitable parameterizations or clusterings requires pre-processing that is difficult, error-prone, and sensitive to the quality of input geometry. The resulting light transport solutions still tend to exhibit discontinuities, necessitating heuristic post-processing before visualization. Due to these problems finite element methods are rarely used in production. The core idea of our approach is to use finite element basis functions induced by hierarchical scattered data approximation techniques. This leads to a mathematically rigorous recipe for meshless finite element illumination computations. As a main advantage, our approach decouples the function spaces used for solving the transport equations from the representation of the scene geometry. The resulting solutions are accurate, exhibit no spurious discontinuities, and can be visualized directly without post-processing, while parameterization, meshing and clustering problems are avoided. The resulting methods are furthermore easy to implement. We demonstrate the power of our framework by describing implementations of hierarchical radiosity, glossy precomputed radiance transfer from distant illumination, and diffuse indirect precomputed transport from local light sources. Moreover, we describe how to directly visualize the solutions on graphics hardware.