The Mongols intended to invade Japan but failed. Genghis Khan had swept down into China in the early 13th century and established the beginnings of the Yuan Dyansty. His grandson Kublai Khan successfully invaded Korea and Burma, but failed twice in attempts to take Japan (1274 and 1281). The so- called Kamikaze-wind (typhoon) saved Japan on one of these occassions and from the Japanese perspective this has been spun into legend saying that Japan was protected by the gods.
As far as being invaded physically and forcefully, I do not think Japan has been taken except during the concluding months of the Pacific War (WWII). Culturally however, Japan often looks upon both the infusion of Korean and Chinese cultures and peoples as invasions of sort, and certainly America has in recent decades been criticized for trying to Americanize (corrupt) Japan, particularly with regard to youth. Japan is currently beset with a generation of youth who are more violent and less moral than any generation in history, and America and Hollywood are often blamed for corrupting (invading if you will) Japan.