Treatment for most persistent viral infections consists of palliative drug options rather than curative approaches. This is often because long-lasting viral DNA in infected cells is not affected by current antivirals, providing a source for viral persistence and reactivation. Targeting latent viral DNA itself could therefore provide a basis for novel curative strategies. DNA cleavage enzymes can be used to induce targeted mutagenesis of specific genes, including those of exogenous viruses. Although initial in vitro and even in vivo studies have been carried out using DNA cleavage enzymes targeting various viruses, many questions still remain concerning the feasibility of these strategies as they transition into preclinical research. Here, we review the most recent findings on DNA cleavage enzymes for human viral infections, consider the most relevant animal models for several human viral infections, and address issues regarding safety and enzyme delivery. Results from well-designed in vivo studies will ideally provide answers to the most urgent remaining questions, and allow continued progress toward clinical application.