We consider fusing an arbitrary number of multiband, i.e., panchromatic, multispectral, or hyperspectral, images of the same scene. Using the well-known forward observation and linear mixture models, a vector total-variation penalty, and appropriate constraints, we cast this problem as a convex optimization problem. The total-variation penalty helps cope with the potential ill-posedness of the underlying inverse problem by exploiting the prior knowledge that natural images are mostly piecewise smooth in the spatial domain and comprise relatively few archetypical signatures, i.e., endmembers, in the spectral domain. We solve the formulated convex but non-smooth optimization problem using the alternating direction method of multipliers. Our experiments with multiband images constructed from real hyperspectral datasets demonstrate the performance advantages of the proposed algorithm over the existing algorithms, which need to be used in tandem to fuse more than two multiband images.