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Protein-protein interactions (PPIs) are essential to all biological processes and they represent increasingly important therapeutic targets. Here, we present a new method for accurately predicting protein-protein interfaces, understanding their properties, origins and binding to multiple partners. Contrary to machine learning approaches, our method combines in a rational and very straightforward way three sequence- and structure-based descriptors of protein residues: evolutionary conservation, physico-chemical properties and local geometry. The implemented strategy yields very precise predictions for a wide range of protein-protein interfaces and discriminates them from small-molecule binding sites. Beyond its predictive power, the approach permits to dissect interaction surfaces and unravel their complexity. We show how the analysis of the predicted patches can foster new strategies for PPIs modulation and interaction surface redesign. The approach is implemented in JET2, an automated tool based on the Joint Evolutionary Trees (JET) method for sequence-based protein interface prediction. JET2 is freely available at www.lcqb.upmc.fr/JET2.