<p>The advancement of multi-material additive manufacturing has enabled the fabrication of multi-material graded cellular structures. However, designing these structures with optimal performance remains challenging due to the absence of an effective multi-scale, multi-material concurrent topology optimization method. To address this gap, this study proposes a novel multi-scale, multi-material concurrent topology optimization framework for the design of graded cellular structures. First, a multi-material multi-scale projection scheme is developed to establish material assignment correspondence between macro and micro scales during optimization. Second, a dimension-reduced multi-material interpolation model is proposed to enhance computational efficiency and the multi-scale concurrent form of the model is correspondingly presented. Third, an existing connectivity scheme for mono-material cells is modified to ensure connectivity between multi-material cells. Fourth, the multi-material multi-scale sensitivity equations accounting for both heat conduction and structural mechanics are derived for the first time. Finally, the effectiveness, generality and stability of the proposed method are evaluated through comparative case studies. The results demonstrate that the proposed approach holds significant potential for integrated structural and material design, advancing the field of multi-material graded cellular structures.</p>