We introduce a general formulation, called non-negative graph embedding, for non-negative data decomposition by integrating the characteristics of both intrinsic and penalty graphs . In the past, such a decomposition was obtained mostly in an unsupervised manner, such as Non-negative Matrix Factorization (NMF) and its variants, and hence unnecessary to be powerful at classification. In this work, the non-negative data decomposition is studied in a unified way applicable for both unsupervised and supervised/semi-supervised configurations. The ultimate data decomposition is separated into two parts, which separatively preserve the similarities measured by the intrinsic and penalty graphs, and together minimize the data reconstruction error. An iterative procedure is derived for such a purpose, and the algorithmic non-negativity is guaranteed by the non-negative property of the inverse of any M-matrix. Extensive experiments compared with NMF and conventional solutions for graph embedding demonstrate the algorithmic properties in sparsity, classification power, and robustness to image occlusions.