The RNA-programmable DNA-endonuclease Cas9 is widely used for genome engineering, where a high degree of specificity is required. To investigate which features of Cas9 determine the sensitivity to mismatches along the target DNA, we performed in vitro biochemical assays and bacterial survival assays in Escherichia coli. We demonstrate that arginines in the Cas9 bridge helix influence guide RNA, and target DNA binding and cleavage. They cluster in two groups that either increase or decrease the Cas9 sensitivity to mismatches. We show that the bridge helix is essential for R-loop formation and that R63 and R66 reduce Cas9 specificity by stabilizing the R-loop in the presence of mismatches. Additionally, we identify Q768 that reduces sensitivity of Cas9 to protospacer adjacent motif-distal mismatches. The Cas9_R63A/Q768A variant showed increased specificity in human cells. Our results provide a firm basis for function- and structure-guided mutagenesis to increase Cas9 specificity for genome engineering.