Modern history of East Asian martial arts
Taekwondo was developed from karate for use in the Republic of Korea Armed Forces from 1954. For this purpose, karate terminology needed to be "nationalized". The name "taekwondo" is due to shotokan practitioner and army general Choi Hong Hi. After the Korean War, Taekwondo was further developed into a purely sportive discipline, since 1973 overseen by the World Taekwondo Federation. 
During 1964-1966, Choi Hong Hi continued to promote taekwondo, working to put together a world-class exhibition team. Taekwondo's emphasis on jumping and flying kicks dates to this period. While taekwondo was based off karate, the structural changes made in the 1950s and 1960s were significant, and it cannot now be considered a branch of karate, but is a standalone modern martial art. Taekwondo was presented at the Summer Olympics as a demonstration sport in 1988 and 1992, and became an official Olympic discipline in 2000.
Martial arts
As Western influence grew in Asia a greater number of military personnel spent time in China, Japan and South Korea during World War II and the Korean War and were exposed to local fighting styles. Jujutsu, judo and karate first became popular among the mainstream from the 1950s-60s. Due in part to Asian and Hollywood martial arts movies, most modern American martial arts are either Asian-derived or Asian influenced.
The term kickboxing (キックボクシング) was created by the Japanese boxing promoter Osamu Noguchi for a variant of muay Thai and karate that he created in the 1950s. American kickboxing was developed in the 1970s, as a combination of boxing and karate. Taekwondo was developed in the context of the Korean War in the 1950s.