Although the earliest evidence of martial arts comes from ancient China, the true roots of martial arts is difficult to reconstruct. Inherent patterns of human aggression which inspire practice of mock combat (in particular wrestling) and optimization of serious close combat as cultural universals are doubtlessly inherited from the pre-human stage, and were made into an "art" from the earliest emergence of that concept. Indeed, many universals of martial art are fixed by the specifics of human physiology and not dependent on a specific tradition or era.
Specific martial arts traditions become identifiable in Classical Antiquity, with disciplines such as Shuai Jiao, Greek wrestling or those described in the Indian epics or the Spring and Autumn Annals of China. In modern times, these traditional roots were often used to legitimize invented traditions which were in fact quite new.

Early Histoty:

The earliest evidence for specifics of martial arts as practiced in the past comes from depictions of fights, both in figurative art and in early literature, besides analysis of archaeological evidence, especially of weaponry.

Wrestling is a human universal, and is also observed in other great apes, especially in juveniles.
The spear has been in use since the Lower Paleolithic and retained its central importance well into the 2nd millennium AD. The bow appears in the Upper Paleolithic and is likewise only gradually replaced by the crossbow, and eventually firearms, in the Common Era. True bladed weapons appear in the Neolithic with the stone axe, and diversify in shape in the course of the Bronze Age (khopesh/kopis, sword, dagger)
Some early examples are the depiction of wrestling techniques in a tomb of the Middle Kingdom of Egypt at Beni Hasan (c. 2000 BC). In ancient China, the Yellow Emperor Huangdi (2698 BC) introduced some of the earliest fighting systems to China. The Yellow Emperor is described as a famous general who, before becoming China’s leader, wrote lengthy treatises on medicine, astrology and the martial arts. The earliest documented form was called Shuai Jiao, a forerunner to the modern art of Chinese Wrestling.

Minoan youths boxing, reconstruction of a Knossos fresco (1500 BC). Earliest evidence for use of gloves.[dubious – discuss]
Literary descriptions of combat begin in the 2nd millennium BC, with cursory mention of weaponry and combat in texts like the Gilgamesh epic or the Rigveda. Detailed description of Late Bronze Age to Early Iron Age hand-to-hand combat with spear, sword and shield are found in the Iliad (c. 8th century BC). Pictorial representations of fist fighting in Minoan civilization date to the 2nd millennium BC.

China:

The genesis of Chinese martial arts has been attributed to the need for self-defense, hunting techniques and military training in ancient China. Hand-to-hand combat and weapons practice were important in training ancient Chinese soldiers.

Detailed knowledge about the state and development of Chinese martial arts becomes available from the Nanjing decade (1928–1937), as the Central Guoshu Institute established by the Kuomintang regime made an effort to compile an encyclopedic survey of martial arts schools. Since the 1950s, the People's Republic of China has organized Chinese martial arts as an exhibition and full-contact sport under the heading of Wushu.

Japan:

The historical origin of Japanese martial arts can be found in the warrior traditions of the samurai and the caste system that restricted the use of weapons by members of the non-warrior classes. Originally, samurai were expected to be proficient in many weapons, as well as unarmed combat, and attain the highest possible mastery of combat skills, for the purpose of glorifying either themselves or their liege. A large number of schools evolved to teach these skills with those existing before the Meiji Restoration classed as Koryū (古流) or old stream. Over time there was a trend away from the traditional purpose to a philosophy of coupling spiritual goals with the striving to perfect their martial skills.

Korea:

Taekkyeon is the Traditional Martial Art of Korea. Taekkyeon's roots date back to the Three Kingdoms of Korea, 57BCE-668CE, at that time Subak was the name for armwork, Taekkyeon was the name for legwork. Subak was the original name for all movements and shares the same origins as its Chinese equivalent Shoubo. Over time Taekkyeon became the single name for all techniques. Taekkyeon was the traditional military martial art focusing on up-right fighting: footwork, kicks, strikes, blocks, throws and rhythm. Its fundamentals are core to integrated advance movements and weapons use.

Ssireum is the Traditional Wrestling Art of Korea. Gakjeochong (각저총:角抵塚) murals show that wrestling in Korea dates back as early as the pre-Three Kingdom era. The Book of Later Han, a Chinese document that was written either before or early in the history of the Three Kingdoms also has records of Korean wrestling[citation needed]. Ssireum first gained widespread popularity during the Joseon Dynasty (1392-1910).

India:

The classical Sanskrit epics contain accounts of combat, describing warriors such as Bhima. The Mahabharata describes a prolonged battle between Arjuna and Karna using bows, swords, trees and rocks, and fists. Another unarmed battle in the Mahabharata describes two fighters boxing with clenched fists and fighting with kicks, finger strikes, knee strikes and headbutts. Other boxing fights are also described in Mahabharata and Ramayana.

The word "kalari" is mentioned in Sangam literature from the 2nd century BC. The Akananuru and Purananuru describe the martial arts of ancient Tamilakkam, including forms of one-to-one combat, and the use of spears, swords, shields, bows and silambam.
A martial art called Vajra Mushti is mentioned in Indian sources of the early centuries AD. Indian military accounts of the Gupta Empire (c. 240-480) identified over 130 different classes of weapons.[citation needed] The Kama Sutra written by Vātsyāyana at the time suggested that women should regularly "practice with sword, single-stick, quarterstaff, and bow and arrow."
Around 630, King Narasimhavarman of the Pallava dynasty commissioned dozens of granite sculptures showing unarmed fighters disarming armed opponents. These may have shown an early form of Varma Adi, a martial art that allowed kicking, kneeing, elbowing, and punching to the head and chest, but prohibited blows below the waist.
Martial arts were not exclusive to the Kshatriya warrior caste, though they used the arts more extensively. The 8th century text Kuvalaymala by Udyotanasuri recorded martial arts being taught at salad and ghatika educational institutions, where Brahmin students from throughout the subcontinent (particularly from South India, Rajasthan and Bengal) "were learning and practicing archery, fighting with sword and shield, with daggers, sticks, lances, and with fists, and in duels (niuddham)."
The earliest extant manual of Indian marital arts is contained as chapters 248 to 251 in the Agni Purana (c. 8th–11th century), giving an account of dhanurveda in a total of 104 shlokas. These verses describe how to improve a warrior's individual prowess and kill enemies using various different methods in warfare, whether a warrior went to war in chariots, elephants, horses, or on foot. Foot methods were subdivided into armed combat and unarmed combat.[17] The former included the bow and arrow, the sword, spear, noose, armour, iron dart, club, battle axe, discus, and the trident. The latter included wrestling, knee strikes, and punching and kicking methods.