Radial basis functions have been used to construct meshfree numerical methods for interpolation and for solving partial differential equations. Recently, a localized basis of radial basis functions has been developed on the sphere. In this dissertation, we investigate applying localized kernel bases for interpolation, approximation, and for novel discretization methods for numerically solving partial differential equations and integral equations. We investigate methods for partial differential equations on spheres using newly explored bases constructed from radial basis functions and associated quadrature methods. We explore applications of radial basis functions to anisotropic nonlocal diffusion problems and we develop theoretical frameworks for these methods.