Human immunodeficiency virus type 1 (HIV-1), hepatitis B virus (HBV), and herpes simplex virus (HSV) have been incurable to date because effective antiviral therapies target only replicating viruses and do not eradicate latently integrated or nonreplicating episomal viral genomes. Endonucleases that can target and cleave critical regions within latent viral genomes are currently in development. These enzymes are being engineered with high specificity such that off-target binding of cellular DNA will be absent or minimal. Imprecise nonhomologous-end-joining (NHEJ) DNA repair following repeated cleavage at the same critical site may permanently disrupt translation of essential viral proteins. We discuss the benefits and drawbacks of three types of DNA cleavage enzymes (zinc finger endonucleases, transcription activator-like [TAL] effector nucleases [TALENs], and homing endonucleases [also called meganucleases]), the development of delivery vectors for these enzymes, and potential obstacles for successful treatment of chronic viral infections. We then review issues regarding persistence of HIV-1, HBV, and HSV that are relevant to eradication with genome-altering approaches.