Korean Martial Arts started over two thousand years ago with a form of Wrestling called Ssireum. Two thousand years later another form of Korean-Martial Art, Tae Kwon Do is the most widely-practiced Martial Art in the world.
In between those two thousand years there isn't a whole lot known about the ancient Martial Arts of Korea, because of the lack of written history about it.
In Korean Martial Arts there are some references to Subak (Empty Hand Fighting) that can be found in records from the Goruryeo dynasty (37 B.C-668 A.D) through the Joseon dynasty (1392-1910). Subak was a style that favored Kicking over Punching, and Stand up Fighting over Grappling. Probably because Subak originated in the mountainous terrain of Goruryeo and the people's legs were relatively more powerful than their upper bodies.
So for the last two thousand years at least, Koreans have been Kicking Butt, literally, with a heavy emphasis on Kicking. It's also recorded that in the 13th century generals in the three kingdoms of Korea-Koguro, Paekchae, and Silla trained in Martial Arts, and that there were Martial Arts contests between the kingdoms. But there were no descriptions of what type of Ancient Martial Arts they trained in or the techniques that were used.
It is known that the majority of Korean-Martial Arts were taught by teachers, and some by Martial Arts manuals. Ancient texts, wall paintings, and sculptures depict people in Martial Arts fighting stances, that are facing off with people in similar stances, the same fighting stances that Martial Artists use today. It was written that the Chinese regarded the Koryo Gi (techniques of Korea) and Yoo-Kyu (a type of Wrestling) as a powerful and superb form of Ancient Martial Arts.
Among the 3 kingdoms the Silla Dynasty (668-935 A.D) was the most famous for developing Martial Arts. An organization known as Hwarang-Do (Way of the Flowery Youth) was composed of young men selected from the upper echelon of the kingdom's youth. They traversed the countrie's mountains, familiarizing themselves with the geography and training in Martial Arts.
They were tough fighters, and were forbidden to retreat in battle. This Hwang-Do was instrumental in unifying the Korean peninsula. Most of the Korean-Martial Arts can trace their spiritual and technical heritage from this group of ancient Martial Artists.
During the Koryo dynasty Buddhism was made the state religion.
Buddhism was against the taking of life and so the common people lost interest in Martial Arts.
But the King of Koryo continued to practice Martial Arts in private. This was the beginning of the Royal Court Martial Arts, and the common people weren't allowed to practice these Arts. These Royal Court Martial Arts were Martial Arts that had been handed down for generations, and only the techniques that were considered the best ones had been kept.
After the dissolution of the Korean Royal Court and the Japanese occupation in 1910 almost all aspects of Korean culture were suppressed by the Japanese, including the teaching of Korean Martial Arts. Those caught teaching Korean Martial Arts were severely punished and many of the best instructors went into hiding.
During the occupation, Koreans were taught Japanese Martial Arts, such as Karate and Judo.
Some of the original Korean Martial Arts survived during the occupation due to the secret teaching of these Arts. After the occupation ancient books like the Muyedobongji (an illustrated manual of Korean Ancient Martial Arts techniques from 1790) influenced the development of modern Korean Fighting Arts. Korean Martial Arts like Tae Kwon Do, were influenced by Okinawan Karate.
Hong Hi Choi, a key figure in the development of Tae Kwon Do, studied Shotokan Karate during the Japanese occupation. And Hwang Kee developed modern Tang Soo Do from the traditional Korean Martial Arts of Taekkyeon, and Subak, combined with Chinese, and Okinawan Martial Arts.
Hapkido had its roots in the Japanese Martial Art of Daito-Ryu Aiki Jui-jitsu, with its Throwing and Joint Locking techniques. Later adding an arsenal of Kicking techniques to the menu. But each differed from the Martial Arts that they were derived from, and became purely Korean with an emphasis on the signature technique of Korean Martial Arts...Kicks.