"Nonnegative matrix factorization (NMF) is a commonly used low-rank model for identifying latent features in nonnegative data. It became a standard tool in applications such as blind source separation, recommender systems, topic modeling, or hyperspectral unmixing. Essentially, NMF consists in finding a few meaningful features such that the data points can be approximated as linear combinations of those features. NMF is generally a difficult problem to solve, since it is both NP-hard and ill-posed (meaning there is no unique solution). However, under the separability assumption, it becomes tractable and well-posed. The separability assumption states that for every feature there is at least one pure data point, that is a data point composed solely of that feature. This is known as the 'pure-pixel' assumption in hyperspectral unmixing.In this presentation I will first provide an overview of separable NMF, that is the family of NMF models and algorithms leveraging the separability assumption. I will then detail recent contributions, notably (i) an extension of this model with sparsity constraints that brings interesting identifiability results; and (ii) new algorithms using the fact that, when the separability assumption holds, then there are often more than one pure data point. I will illustrate the models and methods presented with applications in hyperspectral unmixing. "