Here is a list of martial arts or fighting styles that are incorporated in MMA:

Boxing:

Boxing (sometimes known as pugilism, Anglais boxing, or English boxing) is a combat sport in which two participants (generally) of similar weight fight each other with their fists. Boxing today is conducted in a regulated way, typically in a series of one to three-minute intervals called rounds. Victory is achieved if the opponent is knocked down and unable to get up before the referee counts to ten (a Knockout, or KO), if the opponent is deemed too injured to continue, or if the opponent is knocked down three times in one round (a Technical Knockout, or TKO). If there is no stoppage of the fight before an agreed number of rounds, a winner is determined either by the referee's decision or by judges' scorecards.

Although fighting with fists comes naturally to people, the ancient Greeks were the first to make a sport of it, by giving rules and staging tournaments with professionals. The birth hour of boxing as a sport may mark its allowance as an Olympic game as early as 688 BC. Modern boxing evolved in Europe, particularly Great Britain.

In some countries with their own fighting sports, the sport is referred to as "English Boxing" (e.g. in France to contrast with French Boxing). There are numerous different forms of boxing practiced across the world.

Brazilian Jiu-Jitsu:

Brazilian Jiu-Jitsu (BJJ) is a martial art and combat sport that focuses on grappling and especially ground fighting with the goal of gaining a dominant position and using joint-locks and chokeholds to force an opponent to submit. The art was based on early 20th century Kodokan Judo, which was itself then a recently-developed system (founded in 1882), based on multiple schools (or Ryu) of Japanese Jujutsu.

It promotes the principle that a smaller, weaker person using leverage and proper technique can successfully defend themselves against a bigger, stronger assailant. BJJ can be trained for self defense, sport grappling tournaments (gi and no-gi) and mixed martial arts (MMA) competition. Sparring (commonly referred to as 'rolling') and live drilling play a major role in training, and a premium is placed on performance, especially in competition.

Kickboxing:

Kickboxing refers to the sport of using martial-arts-style kicks and western-boxing-style punches to defeat an opponent in a similar way to that of standard boxing. Kickboxing is a standing sport and does not allow continuation of the fight once a combatant has reached the ground.

Kickboxing is often practised for general fitness, or as a full-contact sport. In the full-contact sport the male boxers are bare chested wearing kickboxing trousers and protective gear including: mouth-guard, hand-wraps, 10oz boxing gloves, groin-guard, shin-pads, kick-boots, and optional protective helmet (usually for those under 18.) The female boxers will wear a tank top in addition to the male clothing/protective gear.

Kickboxing is often confused with Muay Thai, also known as Thai Boxing. The two sports are similar, however, in Thai Boxing, kicks below the belt are allowed, as are strikes with knees and elbows.

Arts labelled kickboxing include:

• Adithada (Indian kickboxing) — A form of kickboxing that uses knee, elbow and forehead strikes
• Lethwei (Burmese kickboxing) — Traditional Burmese martial arts of which has now grown into a popular kickboxing event with strong emphasis on knee, elbow strikes and head butt. Any part of the body may be used to strike and be struck. It us also known as Bando kickboxing.
• Pradal Serey (Khmer kickboxing) — A possible predecessor of Muay Thai
• Muay Thai (Thai boxing) — Traditional Thai martial art of which has now grown into a popular kickboxing event with strong emphasis on knee and elbow strikes
• Japanese kickboxing — Similar to Muay Thai, but different point system is taken (e.g. K-1).
• American kickboxing — Similar to Japanese kickboxing and Full contact karate, but different point system is taken
• Full Contact Karate — Most of the time padding and in some cases body armor is used and is the applicable component of karate like many other styles which also include routines training.
• Savate (French kickboxing) — Allows the use of shoes
• Sanshou/Sanda (Chinese kickboxing) — The applicable component of wushu/kung fu of which Takedowns and throws are legal in competition as well as all other sorts of striking (use of arms and legs).
• Shoot boxing — A Japanese form of kickboxing which allows throwing and submission while standing similar to San Shou
• Yaw-Yan (Filipino kickboxing) — Sayaw ng Kamatayan (Dance of Death) is the proper name for Yaw-Yan, a Filipino martial art developed by Napoleon Fernandez. The art resembles Muay Thai in a sense, but differs in the hip torquing motion as well as downward-cutting of its kicks.

There are many additional derivatives of these forms, as well as combined styles which have been used in specific competitions (e.g. K-1). The rules of 'kickboxing' also vary between these different styles.

Sambo:

Sambo (Russian: ???) -- (also called Sombo or Cambo and sometimes written in all-caps) is a modern martial art, combat sport and self-defense system developed in the former Soviet Union, and recognized as an official sport by the USSR All-Union Sports Committee in 1938, presented by Anatoly Kharlampiev.

The word CAMBO(SAMBO) is an acronym of ??? ? ?? (SAMozashchita Bez Oruzhiya) meaning "self-defense without a weapon" in Russian. Sambo has its roots in Japanese judo and traditional folk styles of wrestling such as Armenian Koch, Georgian Chidaoba, Moldovan Trînt, Uzbek Kurash, Mongolian Khapsagay and Azerbaijani Gulesh.

Wrestling:

Wrestling is the act of physical engagement between two unarmed persons, in which each wrestler strives to get an advantage over or control of their opponent. Physical techniques which embody the style of wrestling are clinching, holding, locking, and leverage. Avoiding techniques likely to lead to serious injury, wrestling has aspects of ritual fighting, but its basic principles are closely related to those of military hand-to-hand combat or self-defence systems. Many styles of wrestling are known all over the world and have long histories, and sport wrestling (particularly amateur wrestling) has been an Olympic sport for over 100 years.