From Wikipedia (Martial Arts):
Martial arts or fighting arts are systems of codified practices and traditions of Combat. Martial arts all have a very similar objective: defend oneself or others from physical threat. In addition, some martial arts are linked to beliefs such as Hinduism, Buddhism, Daoism, Confucianism or Shinto while others follow a particular code of honor. Martial arts are considered as both an art and a science. Many arts are also practiced competitively, most commonly as combat sports, but may also take the form of dance.
From Wikipedia (Mixed Martial Arts):
Mixed martial arts (MMA) is a full contact combat sport that allows a wide variety of fighting techniques and skills, from a mixture of martial arts traditions and non-traditions, to be used in competitions. The rules allow the use of both striking as well as grappling techniques, both while standing and on the ground. Such competitions allow martial artists of different backgrounds to compete.