Motivated by the pressing need to characterize protein-DNA and protein-RNA interactions on large scale, we review a comprehensive set of 30 computational methods for high-throughput prediction of RNA- or DNA-binding residues from protein sequences. We summarize these predictors from several significant perspectives including their design, outputs and availability. We perform empirical assessment of methods that offer web servers using a new benchmark data set characterized by a more complete annotation that includes binding residues transferred from the same or similar proteins. We show that predictors of DNA-binding (RNA-binding) residues offer relatively strong predictive performance but they are unable to properly separate DNA- from RNA-binding residues. We design and empirically assess several types of consensuses and demonstrate that machine learning (ML)-based approaches provide improved predictive performance when compared with the individual predictors of DNA-binding residues or RNA-binding residues. We also formulate and execute first-of-its-kind study that targets combined prediction of DNA- and RNA-binding residues. We design and test three types of consensuses for this prediction and conclude that this novel approach that relies on ML design provides better predictive quality than individual predictors when tested on prediction of DNA- and RNA-binding residues individually. It also substantially improves discrimination between these two types of nucleic acids. Our results suggest that development of a new generation of predictors would benefit from using training data sets that combine both RNA- and DNA-binding proteins, designing new inputs that specifically target either DNA- or RNA-binding residues and pursuing combined prediction of DNA- and RNA-binding residues.