Understanding of molecular mechanisms that govern protein-protein interactions and accurate modeling of protein-protein docking rely on accurate identification and prediction of protein-binding partners and protein-binding residues. We review over 40 methods that predict protein-protein interactions from protein sequences including methods that predict interacting protein pairs, protein-binding residues for a pair of interacting sequences and protein-binding residues in a single protein chain. We focus on the latter methods that provide residue-level annotations and that can be broadly applied to all protein sequences. We compare their architectures, inputs and outputs, and we discuss aspects related to their assessment and availability. We also perform first-of-its-kind comprehensive empirical comparison of representative predictors of protein-binding residues using a novel and high-quality benchmark data set. We show that the selected predictors accurately discriminate protein-binding and non-binding residues and that newer methods outperform older designs. However, these methods are unable to accurately separate residues that bind other molecules, such as DNA, RNA and small ligands, from the protein-binding residues. This cross-prediction, defined as the incorrect prediction of nucleic-acid- and small-ligand-binding residues as protein binding, is substantial for all evaluated methods and is not driven by the proximity to the native protein-binding residues. We discuss reasons for this drawback and we offer several recommendations. In particular, we postulate the need for a new generation of more accurate predictors and data sets, inclusion of a comprehensive assessment of the cross-predictions in future studies and higher standards of availability of the published methods.