In　this　paper,　we　use　the　finite　element　method　(FEM)　to　solve　the　time-space　fractional　Bloch-Torrey　equation　on　irregular　domains　in　R3.　Based　on　linear　Lagrange　basis　functions,　a　space　semi-discrete　FEM　scheme　is　given.　By　adopting　the　L2−1σ　approximation　for　the　Caputo　fractional　derivative,　a　fully　discrete　scheme　is　presented.　Furthermore,　we　provide　the　details　on　how　to　implement　our　FEM　for　the　space　fractional　Bloch-Torrey　equation.　Also,　the　stability　and　convergence　of　the　fully　discrete　scheme　is　investigated.　The　error　estimations　with　respect　to　the　L2　and　energy　norms　are　given.　In　addition,　some　numerical　examples　are　presented　to　verify　the　efficiency　of　our　method.　©　2020　Elsevier　Inc.