In 1274 and 1281 Kublai Khan decided to attack Japan at Hakata Bay. During this period of time nobody in the world had seen a fighting force such as the Mongol hoard. Everyone that had opposed them lost. During the Mongol's first invasion in 1274 the Mongols were easily defeating the Japanese. The Japanese had basically been fighting amongst themselves so they were not familiar with the military tactics of outsiders. Samurai would step out, announce themselves and their linage before battle. However the Mongols were not aware of this tradition, nor did they care. When a samurai stepped out to announce himself the Mongols would shoot poison tipped arrows at the sitting duck. The Mongols also had superior technology now that they had conquered China. They employed weapons such as bombs, which the Japanese did not have. In any case the battle was going poorly for the Japanese when a Typhoon came and wiped out the Mongol fleet saving Japan from the Mongols.
In 1281 Kublai Khan attacked Japan again at Hakata Bay, believing the first loss was just bad luck. The second time the Japanese were more prepared. They built a wall along Hakata Bay to slow down the Mongol horsemen and raiders. The Japanese were not deep sea farers during this period however they had prepared small boats that could row out to the Mongol Ships. There were two forces for this attack and they were to attack together. However one of the forces refused to wait for the other and attacked. The fighting was fierce with the Japanese using the small ships to row out to the ships under the cover of night. There the samurai could use their strength, hand to hand combat, against the Mongols who were horsemen, not sea faring fighters. Many of the Chinese soldiers in the Mongol fleet were there because they were not truly loyal to Kublai Khan; these men had previously had good relations with Japan and had no true desire to wage war against them. The fighting was fierce and the Japanese put up a valiant effort, but eventually there were just too many Mongols.
Again when things looked very bad for the Japanese another typhoon came and sunk the Mongol fleet. With the fleet destroyed victory went to the Japanese again. The Japanese believed that they were under divine protection and thus called these typhoons Kamikaze, Divine Wind. The defeats in Japan was a huge hit to the Mongol treasury and is widely considered to be what led to the downfall of Kublai Khans latter years.
On 10/11/08 I traveled to Hakata Bay. The city is now known as Fukuoka. I wouldn't recommend this site unless it really interests you. It interested me though, and since I had a JR pass I just took the bullet train from Osaka to Hakata, visit the bay, and then took the bullet train back to my hostel in Osaka.
Here are some pictures of what's left of the Wall at Hakata Bay and a few pictures of the bay itself.
Links with information on the Mongol invasion of Japan