Description
Brazilian Jiu-Jitsu is primarily a ground-fighting art. Most techniques involve both fighters on the mat. There is a heavy emphasis on positional strategy such as which fighter is on top and where each person's legs are. Positions are stable situations from which a large variety of techniques are available to both fighters.
Specific techniques are designed to improve one's position, to "pass the guard" for example by going from being "in the guard" to getting around the opponent's legs, resulting in side control. Techniques are also taught to defeat your opponent by way of submission. Most submissions are either chokes (cutting off the blood supply to the brain) or arm locks (hyper extending the elbow, or twisting the shoulder).
Most training has students wearing a heavy gi or kimono on a floor with padded mats. A typical class involves 30 minutes of warm ups and conditioning, 30 minutes of technique practice with a willing partner, and 30 minutes of free sparring training, against an opponent of equal skill who attempts to submit you.
Class in action (click to enlarge)

