Point-based surface processing has developed into an attractive alternative to mesh-based processing tools for a number of geometric modeling applications. By working with point clouds directly, processing is based on the raw data and its underlying geometry rather than any arbitrary intermediate representations and generally artificial connectivity relations. We extend this principle into the area of subdivision surfaces by introducing the notion of meshless geometric subdivision. Our approach replaces the role of mesh connectivity with intrinsic point proximity thereby avoiding a number of limitations of mesh-based surface subdivision schemes. Apart from introducing this idea of meshless subdivision, we put forward a first intrinsic meshless subdivision scheme and present a new method for the computation of intrinsic means on Euclidean manifolds.