In this paper,a linear implicit L1-Legendre Galerkin Chebyshev collocation method for the generalized time-and space-fractional Burgers equation is developed.A linear implicit finite difference scheme based on the L1-algorithm for the Caputo fractional derivative is proposed for temporal discretization.And the Legendre Galerkin Chebyshev collocation method,based on the Legendre-Galerkin variational form,but the nonlinear term and the right-hand term are treated by Chebyshev-Gauss interpolation,is proposed for spatial discretization.Rigorous stability and convergence analysis are developed.Numerical examples are shown to demonstrate the accuracy,stability and effectiveness of the method.