Consensus clustering and semi-supervised clustering are important extensions of the standard clustering paradigm. Consensus clustering (also known as aggregation of clustering) can improve clustering robustness, deal with distributed and heterogeneous data sources and make use of multiple clustering criteria. Semi-supervised clustering can integrate various forms of background knowledge into clustering. In this paper, we show how consensus and semi-supervised clustering can be formulated within the framework of nonnegative matrix factorization (NMF). We show that this framework yields NMF-based algorithms that are: (1) extremely simple to implement; (2) provably correct and provably convergent. We conduct a wide range of comparative experiments that demonstrate the effectiveness of this NMF-based approach.