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The actual connotation of randori depends on the martial art it is used in. In judo, jujitsu, and Shodokan aikido, among others, it most often refers to one-on-one sparring where partners attempt to resist and counter each other's techniques. In other styles of aikido, in particular Aikikai, it refers to a form of practice in which a designated aikidoka defends against multiple attackers in quick succession without knowing how they will attack or in what order.
The term is used in aikido, judo, and Brazilian jiu-jitsu dojos outside Japan. In Japan, this form of practice is called taninzu-gake (多人数掛け), which literally means multiple attackers.
The term was described by Jigoro Kano, the founder of Judo, in a speech at the 1932 Los Angeles Olympic Games: "Randori, meaning "free exercise", is practiced under conditions of actual contest. It includes throwing, choking, holding the opponent down, and bending or twisting of the arms. The two combatants may use whatever methods they like provided they do not hurt each other and obey the rules of Judo concerning etiquette, which are essential to its proper working." 
In Tenshin Aikido
In Steven Seagal's Tenshin Aikido Federation (affiliated with the Aikikai), randori is different from that of Aikikai, in that the attackers can do anything to the defender (e.g. punch, grab, kick, etc.), and the randori continues on the ground until a pin.
In kendo, jigeiko means "friendly" free combat, as in competition, but without counting points.
Although in karate the word kumite is usually reserved for sparring, some schools also employ the term randori with regard to "mock-combat" in which both karateka move very fast, parrying and attempting acts of extreme violence with all four limbs (including knees, elbows, etc.), yet only ever making the slightest contact. Total control of the body is necessary, and therefore only the senior grades can typically practice randori. In these schools, the distinction between randori and kumite is that in randori, the action is uninterrupted when a successful technique is applied. (Also known as ju kumite or soft sparring.)
Randori is also practiced in Bujinkan ninjutsu and usually represented to the practitioner when he reaches the "Shodan" level. In ninjutsu, randori puts the practitioner in a position where he is armed or unarmed and is attacked by multiple attackers.
- Original text of this speech available at The Judo Information Site at http://judoinfo.com/kano1.htm
- Ohlenkamp, Neil (16 May 2018). "Black Belt Judo". New Holland – via Google Books.
- Tello, Rodolfo (1 August 2016). "Judo: Seven Steps to Black Belt (An Introductory Guide for Beginners)". Amakella Publishing – via Google Books.