An international organisation of evil (as it is repeatedly called) bent on world domination. KAOS was founded in 1904 in Bucharest [Episode #38: "Hoo Done It"] but is a Delaware corporation for tax purposes [Episode #59: "A Man Called Smart, Part 2"].
Their leader is known as Mr. Big, although this appears to be a descriptive term rather than an actual title, and at least two different men have been referred to as such [Episodes #1: "Mr. Big" and #26: "Hubert's Unfinished Symphony"].
Only once has an individual been seen identified as Chief of KAOS, equivalent to the Chief of CONTROL. This was when the two chiefs attended the funeral of Yohan, a double agent who worked for both KAOS and CONTROL [Episode #85: "Die, Spy"].
Vice President of Public Relations and Terror is Ludwig Siegfried.
For the most part, however, KAOS appears to operate as a series of autonomous cells, each with its own leader and agenda. Even as one cell is destroyed, another is ready to carry out its plan, making the organization very resilient. (Privately, Maxwell Smart is grateful for this state of affairs, noting that if KAOS was ever destroyed there would be no need for CONTROL, thus putting him out of a job [Episode #35: "Maxwell Smart, Alias Jimmy Ballantine"].
Although not spelled out there are clear parallels between KAOS and the communist bogey of the 1960s, with many KAOS agents hailing from Eastern European backgrounds or, as Max calls it, "the not-so-free world" [Episode #82: "Spy, Spy, Birdie"]. As with their real-world counterparts there is a rift between the European and Asian branches. This is graphically demonstrated when a KAOS agent posing as the beauty contestant Miss Formosa shoots dead two agents from Transmania. Obviously seeing them as rivals rather than allies, she declares, "Your brand of KAOS is weak and confused; only in my country is there pure KAOS" [Episode #50: "The Girls from KAOS"].
As time passes, KAOS organises itself in a more business-like fashion, even making a profit on its operations [Episode #54: "The Expendable Agent"], and eventually becomes a takeover target; first by mogul Ironhand [Episode #114: "Ironhand"] and later by publisher Nicholas Dimente [TV Movie "Get Smart, Again!"].
In the nineties, the mysterious KAOS Chairwoman takes this to its logical conclusion and acknowledges the end of the Cold War by refocussing KAOS's goal of world domination on economic rather than political means ["Get Smart (1995 series)"]