Hyperspectral unmixing has attracted much attention in recent years. Single sparse unmixing assumes that a pixel in a hyperspectral image consists of a relatively small number of spectral signatures from large, ever-growing, and available spectral libraries. Joint-sparsity (or row-sparsity) model typically enforces all pixels in a neighborhood to share the same set of spectral signatures. The two sparse models are widely used in the literature. In this paper, we propose a joint-sparsity-blocks model for abundance estimation problem. Namely, the abundance matrix of size <inline-formula> <tex-math notation="LaTeX">m\times n </tex-math></inline-formula> is partitioned to have one row block and <inline-formula> <tex-math notation="LaTeX">s </tex-math></inline-formula> column blocks and each column block itself is joint-sparse. It generalizes both the single (i.e., <inline-formula> <tex-math notation="LaTeX">s=n </tex-math></inline-formula>) and the joint (i.e., <inline-formula> <tex-math notation="LaTeX">s=1 </tex-math></inline-formula>) sparsities. Moreover, concatenating the proposed joint-sparsity-blocks structure and low rankness assumption on the abundance coefficients, we develop a new algorithm called joint-sparse-blocks and low-rank unmixing . In particular, for the joint-sparse-blocks regression problem, we develop a two-level reweighting strategy to enhance the sparsity along the rows within each block. Simulated and real-data experiments demonstrate the effectiveness of the proposed algorithm.