In this paper, we devote to the study of high order finite difference schemes for one- and two-dimensional time–space fractional sub-diffusion equations. A fourth order finite difference scheme is invoked for the spatial fractional derivatives, and the L1 approximation is applied to the temporal fractional parts. For the two-dimensional case, an alternating direction implicit scheme based on L1 approximation is proposed. The stability and convergence of the proposed methods are studied. Numerical experiments are performed to verify the effectiveness and accuracy of the proposed difference schemes.