The Department of Extranormal Operations was a security department of the United States of America originally headed by the Metahuman Director Bones. The main purpose of the agency was to do the jobs that the FBI or CIA would not do, mostly involving vigilantes like Batman and those with extranormal or supposed extranormal powers. The agency was mostly secular, not working with other government groups. Director Bones' obsession with the Kane Family, leading to a schism with the Batman Family.
The DEO, much like other agencies, had a paramilitary force. However, this was used sparingly due to Bones' manipulation of superheroes and supervillains. Hence, most of the agents of DEO, despite being trained for combat, served as an intelligence wing.
Created prior to the public reveal of Superman and after the creation of SHADE, the DEO handled many problems within the US that other agencies were either unable to do, or refused to do. Oftentimes this would take the organization into places that were unbelievable. Eventually, Director Bones' obsession with the Kane Family led to him going rogue, being shot in the head by Agent Asaf and sent into a coma. Bones was then replaced by Asaf.