Accurate modification of the 3 billion-base-pair human genome requires tools with exceptional sequence specificity. Here, we describe a general strategy for the design of enzymes that target a single site within the genome. We generated chimeric zinc finger recombinases with cooperative DNA-binding and catalytic specificities that integrate transgenes with >98% accuracy into the human genome. These modular recombinases can be reprogrammed: New combinations of zinc finger domains and serine recombinase catalytic domains generate novel enzymes with distinct substrate sequence specificities. Because of their accuracy and versatility, the recombinases/integrases reported in this work are suitable for a wide variety of applications in biological research, medicine, and biotechnology where accurate delivery of DNA is desired.