The Mongols were an obscure people who lived in the outer reaches of the Gobi Desert in what is now Outer Mongolia. They spent much of the time arguing and fighting amongst themselves. All that changed with Genghis Khan. The Mongols under his command literally decimated populations in Western Asia and China as they advanced. As a result of all these tactics, the Mongol armies spread across the landscape like wildfire. They marched inexorably south into Chin territory and west into Asia and even Europe.
While Mongol armies spread quickly west, Genghis Khan preceded cautiously in expanding southward, conquering first the northern Tibetan kingdom and later the Chin empire. When he died in 1227, he had just finished conquering the northern city of Beijing. By 1241, the Mongols had conquered all of northern China.
In 1260, Kublai Khan, a grandson of Genghis Khan, became Great Khan. Four years later he relocated his capital from Mongolia to Beijing in northern China, and in 1271 he adopted a Chinese dynastic name, the Yuan. Kublai Khan had decided to become the emperor of China and start a new dynasty; within a few short years, the Mongols had conquered all of southern China.
The Mongols were aloof rulers, not learning the Chinese language, and although many became Buddhists, they didn't really support Buddhism. Meanwhile Chinese culture continued much as before. Nonetheless, the Mongol rulers were very preoccupied with religions. Kublai Khan in particular invited all sorts of faiths to debate at his court. He allowed Nestorian Christians and Roman Catholics to set up missions, as well as Tibetan lamas, Muslims, and Hindus. The Yuan period, in fact, is one of vital cultural transmission between China and the rest of the world. Europe formally met China during the reign of Kublai Khan with the arriuval of Marco Polo, an Italian adventurer, who served as an official in Kublai's court from 1275-1291.
After just over a hundred years, the Yuan Dynasty fell. A peasant, Chu Yuan-chang, led a rebel army against the Yuan. He had lost most of his family in a famine, and had spent part of his life as a monk and then as a bandit leader. He took Beijing in 1368 and the Yuan emperor fled to Shangtu. When he drove the Yuan from Shangtu back to Mongolia, he declared himself the founder of a new dynasty: the Ming (1369-1644).