The Hunting Dogs is an official special military division tasked by the government to hunt down and capture ability users with powerful/dangerous abilities when they begin abusing them and using them for criminal/terrorist actions, making them one of the most powerful military units available to the government.
The Hunting Dogs are also known to be one of the strangest if not the most bizarre group in the military due to the several unique personalities and quirks of its members and its leader's own eccentricities and bizarre habits.
The Hunting Dogs is known to be the most powerful Ability User Task Force within the nation of Japan with a undefeated track record in mission successes and a high attrition rate of survival. Effectively granting them an immense sway over the government, especially to the various law enforcement agencies and groups of the country, as seen where several of its members were able to order around even members of the Special Division for Unusual Powers and overturn their authority when ordered into investigations. 
Due to their immense success, the Hunting Dogs have even garnered international stardom and have even been requested by other countries to assist them in handling Ability related operations and crisis's globally. As seen where their leader Ōchi Fukuchi was noted to have handled numerous high risk supernatural missions and defeated several powerful Ability User that eventually earned him immense international fame.
Due to their skills, the Hunting Dogs have also earned an immense reputation of fear in the Underground of Ability User Crime. With their reputation and viciousness in accomplishing their missions earning them the moniker "Mad Dogs".
In order to join the unit, the candidate must first agree to being surgically altered by the government prior to being accepted into the group, where their bodies would then be physically altered and enhanced to grant them superhuman abilities such as enhanced speed, reflexes, strength, durability and stamina by the groups' Abilities Engineer. 
However these modifications are also known to be the method the government uses to maintain control of the Hunting Dogs. As stated by one of their members, Jōno in his attempts to recruit Kunikida and Taruko when describing the commitment of being a Hunting Dog, the moment the surgery is completed, the physical changes to the potential recruit's bodies would need monthly surgeries to maintain their new bodies that would otherwise break down and self destruct on its own by rotting from the inside out if left alone. This insures that even if the new member isn't actually loyal to the group itself, they have no choice but to comply and stay in the unit or die, broadening the potential candidates to even wanted criminals. 
Aside from the augmentation surgery, Teruko has stated that the duty of a Hunting Dog is to be a "slave" to society and to be the "ultimate violence" above all others in order to preserve the peace and the order of the nation and the world even if it means sacrificing one's own bodies and souls for the people; a Hunting Dog must fight while injured and to the death, further showcasing the commitment and resolve a loyal member must have to be in the unit. 
- Harukawa stated in a magazine interview that the green color motif for the Hunting Dogs was based on Tachihara's green military crop jacket.