As a powerful tool for data representation, deep NMF has attracted much attention in recent years. Current deep NMF builds the multi-layer structure by decomposing either basis matrix or feature matrix into multiple factors, and probably complicates the learning process when data is insufficient or exhibits simple structure. To overcome the limitations, a novel method called Generalized Deep Non-negative Matrix Factorization (GDNMF) is proposed, which generalizes several NMF and deep NMF methods in a unified framework. GDNMF simultaneously performs decomposition on both features and bases, which learns a hierarchical data representation based on multi-level basis. To further improve the latent representation and enhance its flexibility, GDNMF mutually reinforces shallow linear model and deep non-linear model. Moreover, semi-supervised GDNMF is proposed by treating partial label information as soft constraints in the multi-layer structure. An efficient two-phase optimization algorithm is developed, and experiments on five real-world datesets verify its superior performance compared with state-of-the-art methods.