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Source separation models that make use of nonnegativity in their parameters have been gaining increasing popularity in the last few years, spawning a significant number of publications on the topic. Although these techniques are conceptually similar to other matrix decompositions, they are surprisingly more effective in extracting perceptually meaningful sources from complex mixtures. In this article, we will examine the various methodologies and extensions that make up this family of approaches and present them under a unified framework. We will begin with a short description of the basic concepts and in the subsequent sections we will delve in more details and explore some of the latest extensions.