Introduction
The Mongols were a nomadic empire that existed from the 13th to the 14th century. Under the leadership of Genghis Khan, they conquered a vast territory spanning from Eastern Europe to Asia. During their reign, the Mongols had various relationships with other states, which can be classified into three categories: alliances, tributary states, and conquests.

Alliances
1. Mongol-Timurid Alliance: The Mongols formed an alliance with the Timurid Empire, led by Timur, in the late 14th century. This alliance helped both empires maintain their dominance in the region and led to the expansion of their territories.

2. Mongol-Jin Dynasty Alliance: The Mongols established an alliance with the Jin Dynasty to defeat the Western Xia Dynasty. This alliance allowed the Mongols to gain a strategic advantage and eventually conquer the Jin Dynasty.

3. Mongol-Tibetan Alliance: The Mongols formed an alliance with Tibet to secure their southern borders and maintain control over trade routes. This alliance also helped the Mongols spread their influence in the region and establish a strong presence in Tibet.

Tributary States
1. Song Dynasty: The Mongols established a tributary relationship with the Song Dynasty after defeating them in the Battle of Yamen. The Song Dynasty paid tribute to the Mongols and acknowledged their supremacy, allowing the Mongols to control the region.

2. Kievan Rus: The Mongols established a tributary relationship with the Kievan Rus, a federation of East Slavic tribes. The Kievan Rus paid tribute to the Mongols and accepted their rule, allowing the Mongols to control the region and exploit its resources.

3. Abbasid Caliphate: The Mongols established a tributary relationship with the Abbasid Caliphate. The Abbasid Caliphate paid tribute to the Mongols and recognized their authority, allowing the Mongols to control the region and influence Islamic affairs.

Conquests
1. Eastern Europe: The Mongols conquered Eastern Europe, including Russia, Hungary, and Poland. They established the Golden Horde, a Mongol state, in the region and imposed their rule over the conquered territories.

2. Central Asia: The Mongols conquered Central Asia, including the Khwarazmian Empire and the Ilkhanate. They established the Chagatai Khanate, another Mongol state, in the region and expanded their influence in Central Asia.

3. China: The Mongols conquered China and established the Yuan Dynasty. They ruled over China for several decades and implemented various administrative and cultural reforms.

Conclusion
The Mongols had complex relations with other states during their reign. They formed alliances with some states to strengthen their positions and expand their territories. They also established tributary relationships with other states, which allowed them to control the regions and maintain their dominance. Additionally, through conquests, they expanded their empire and established Mongol states in various regions. The Mongols' interactions with other states played a significant role in shaping their empire and influencing the history of the regions they conquered.