In　this　paper,　we　consider　a　Legendre　spectral　method　on　graded　meshes　for　solving　the　two-dimensional　multi-term　time-fractional　diffusion　equation　with　non-smooth　solutions.　The　L1　method　on　graded　meshes　is　developed　in　temporal　direction.　In　the　spatial　direction,　we　choose　the　Legendre　spectral　method.　With　the　help　of　mathematical　induction　method,　the　stability　and　convergence　of　the　above　mentioned　method　are　theoretically　proved.　To　improve　the　computational　efficiency,　we　use　the　fast　algorithm　in　the　numerical　implementation.　Some　numerical　results　are　provided　to　verify　the　theoretical　analysis　and　demonstrate　the　effectiveness　of　our　method.　(C)　2020　Elsevier　Ltd.　All　rights　reserved.