This thesis is concerned with the numerical solution of advection-dominated diffusion problems. There are essentially two key aspects to this work: the derivatives of an a priori error estimate for a semi-Lagrangian mesh-free method using radial basis function interpolation to numerically approximate the first-order linear transport problem; and the design and testing of a semi-Lagrangian mesh-less method to numerically solve the full parabolic advection-diffusion problem, using radial basis function Hermite interpolation. We begin by establishing the theory of radical basis function interpolation, including new results for the stability of interpolation via the class of radial basis functions known as polyharmonic splines, as well as error estimates for interpolation by the same class of function. These results provide us with the necessary tools to prove the a priori error estimate for the semi-Lagrangian advection scheme, given certain assumptions on the smoothness of the solution. We then validate both the scheme and the analysis with a series of numerical experiments. By introducing the concept of Hermite interpolation, we develop and implement a new semi-Lagrangian method for the numerical approximation of advection-dominated diffusion problems, which is validated through two numerical experiments.