Models　based　on　partial　differential　equations　containing　time–space　fractional　derivatives　have　attracted　considerable　interest　in　the　past　decade　because　of　their　ability　to　model　anomalous　transport　phenomena.　These　phenomena　are　strongly　connected　to　the　interactions　within　complex　and　non-homogeneous　media　exhibiting　spatial　heterogeneity.　The　class　of　equations　with　multi-term　time–space　derivatives　of　fractional　orders　has　been　found　to　be　very　useful　in　the　description　of　such　interactions.　This　motivates　the　extension　of　the　classical　Bloch–Torrey　equation　through　the　application　of　the　operators　of　fractional　calculus　to　new　multi-term　time–space　fractional　Bloch–Torrey　equations　with　Riesz　fractional　operators.　In　this　paper,　we　firstly　propose　an　unstructured-mesh　Galerkin　finite　element　method　for　the　two-dimensional　multi-term　time–space　fractional　diffusion　equation　with　Riesz　fractional　operators　on　irregular　convex　domains.　Secondly,　we　rigorously　establish　the　stability　and　convergence　of　the　numerical　scheme.　Thirdly,　we　extend　the　computational　model　to　solve　a　system　of　coupled　two-dimensional　multi-term　time–space　fractional　Bloch–Torrey　equations.　Finally,　some　numerical　results　are　given　to　demonstrate　the　versatility　and　application　of　the　models.　©　2019　Elsevier　Ltd