In this paper, we consider a space-time Riesz---Caputo fractional advection-diffusion equation. The equation is obtained from the standard advection-diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative of order ????(0,1], the first-order and second-order space derivatives by the Riesz fractional derivatives of order β 1???(0,1) and β 2???(1,2], respectively. We present an explicit difference approximation and an implicit difference approximation for the equation with initial and boundary conditions in a finite domain. Using mathematical induction, we prove that the implicit difference approximation is unconditionally stable and convergent, but the explicit difference approximation is conditionally stable and convergent. We also present two solution techniques: a Richardson extrapolation method is used to obtain higher order accuracy and the short-memory principle is used to investigate the effect of the amount of computations. A numerical example is given; the numerical results are in good agreement with theoretical analysis.