In this thesis, we consider a separable preconditioner for linear systems arising from time-space Caputo-Riesz fractional diffusion equations. We give an introduction in Chapter 1. In Chapter 2, we propose a separable preconditioner for linear systems arising from full discretization of time-space Caputo-Riesz fractional diffusion equations. Theoretically, we show that if the diffusion coefficient function is temporalindependent or spatial-independent, singular values of the preconditioned matrix are bounded above and below by positive constants which are independent of discretization parameters. Numerical examples are reported to demonstrate that the performance of the proposed preconditioner is efficient, and is better than other approaches. This thesis ends with a chapter of a brief conclusion.