The radial basis functions (RBF) method is mesh free, easy to implement in any number of dimensions, and spectrally accurate for certain types of radial functions. However, it still has stability and complexity issues, which keep it from being used more widely. In this dissertation, we study problems related to the complexity and stability properties of the RBF method. In particular, we first study the locality property of the RBF's expansion coefficients. We are able to show and quantify how a perturbation in the function value at one node will affect expansion coefficients associated with only the neighboring nodes. This locality property is a key in the development of fast iterative methods (Powell, Faul, etc.), and our study is valuable in a time where the lack of a generally applicable fast algorithm is one of the biggest obstacles that the RBF methodology is facing. We also study the role of the shape parameter on the stability of the method. It has been known for quite some time that certain nice properties (especially high accuracy) are linked with flat RBF interpolants (small values of the shape parameter). However, the lack of stability associated with small shape parameters has led people to believe that computations of the interpolant in this regime of the shape parameter were impossible. Fornberg and Wright developed the first tool (the Contour-Padé algorithm) to get around the instability and thereby disprove the phenomenon described by Schaback as the "uncertainty principle." We present a second algorithm, which we call the RBF-QR algorithm. It is also designed to stably compute interpolants in the case of flat RBFs but is easier to implement and does not have restrictions on the number of nodes. We use this tool to study the role of the shape parameter on the error when interpolating as well as when solving a convection equation on the sphere.