There were two major attempted Mongol invasions of Japan, launched in 1274 and 1281. Both expeditions were ordered by Kublai Khan, founder of China’s Yuan dynasty, after Japan rejected repeated demands to submit to Mongol authority.
The first invasion began in November 1274. A combined Yuan and Korean force attacked Tsushima and Iki before landing in northern Kyushu. Japanese defenders struggled against the invaders’ coordinated tactics, archery, and explosive weapons, but the expedition withdrew after fighting around Hakata Bay. A storm damaged much of the retreating fleet.
Japan prepared extensively for a second attack by building coastal defenses, including a stone wall around Hakata Bay. The much larger 1281 invasion arrived in two main fleets from Korea and southern China. Japanese resistance, nighttime raids, fortified shorelines, and a powerful typhoon forced the attackers to withdraw. The storm later became associated with the term kamikaze, or “divine wind.”
The question asks about invasion attempts, not individual battles or landings. Many engagements occurred during each campaign, but historians conventionally count the campaigns as two invasions.