BACKGROUND: The propensity for off-target activity of Streptococcus pyogenes Cas9 (SpCas9) has been considerably decreased by rationally engineered variants with increased fidelity (eSpCas9; SpCas9-HF1). However, a subset of targets still generate considerable off-target effects. To deal specifically with these targets, we generated new "Highly enhanced Fidelity" nuclease variants (HeFSpCas9s) containing mutations from both eSpCas9 and SpCas9-HF1 and examined these improved nuclease variants side by side to decipher the factors that affect their specificities and to determine the optimal nuclease for applications sensitive to off-target effects. RESULTS: These three increased-fidelity nucleases can routinely be used only with perfectly matching 20-nucleotide-long spacers, a matching 5' G extension being more detrimental to their activities than a mismatching one. HeFSpCas9 exhibit substantially improved specificity for those targets for which eSpCas9 and SpCas9-HF1 have higher off-target propensity. The targets can also be ranked by their cleavability and off-target effects manifested by the increased fidelity nucleases. Furthermore, we show that the mutations in these variants may diminish the cleavage, but not the DNA-binding, of SpCas9s. CONCLUSIONS: No single nuclease variant shows generally superior fidelity; instead, for highest specificity cleavage, each target needs to be matched with an appropriate high-fidelity nuclease. We provide here a framework for generating new nuclease variants for targets that currently have no matching optimal nuclease, and offer a simple means for identifying the optimal nuclease for targets in the absence of accurate target-ranking prediction tools.