Administrative divisions of the Yuan dynasty

The Yuan dynasty was a vast empire founded by Mongol leader Kublai Khan in China. During its existence, its territory was divided into the Central Region (腹裏) governed by the Central Secretariat (Zhongshu Sheng) and places under control of various provinces (行省) or Branch Secretariats (行中書省), as well as the region under the Bureau of Buddhist and Tibetan Affairs (Xuanzheng Yuan). In addition, the Yuan emperors held nominal suzerainty over the western Mongol khanates, but in reality none of them were governed by the Yuan dynasty due to the division of the Mongol Empire.
Overview.
The most important part of the Yuan Empire was the Central Region, which covered the region of the Yuan capital Khanbaliq (Dadu, modern Beijing). The Central Region consisted of present-day Hebei, Shandong, Shanxi, the south-eastern part of present-day Inner Mongolia and the Henan areas to the north of the Yellow River. Since it was considered the most important region of the dynasty, it was directly governed by the Zhongshu Sheng at Khanbaliq; similarly, another top-level administrative department called the Bureau of Buddhist and Tibetan Affairs (or Xuanzheng Yuan) held administrative rule over Tibet.
Branch Secretariats (行中書省), or simply provinces (行省), were provincial-level administrative organizations or institutions subordinated to the Zhongshu Sheng, though they were not exactly provinces in modern sense. There were 11 "regular" provinces in Yuan dynasty.
Below the level of provinces, the largest political division was the circuit (), followed by lù (), fǔ () and zhōu (). These are three kinds of prefecture-like divisions. The lowest political division was the xiàn or counties (). 
Basically, lù is higher than fǔ, and fǔ is higher than zhōu.However, the actual relationship between them could be very complicated. Both lù, fǔ and zhōu could administer counties. Some fǔ and zhōu are directly administered by the province, while some exist inside a lù. A lù usually administers several counties, along with several fǔ and zhōu, and the fǔ or zhōu themselves could also administer their own counties. As a result, it is impossible to exactly define how many tiers of divisions there are under a province.
This government structure at the provincial level was later inherited and modified by the Ming and Qing dynasties.
Special provinces (Branch Secretariats).
These provinces or Branch Secretariats were set up before or during the invasions of certain regions outside the Yuan, such as Japan, Vietnam and Burma. They were usually abolished after the wars.
Region under the Bureau of Buddhist and Tibetan Affairs.
The Bureau of Buddhist and Tibetan Affairs, or Xuanzheng Yuan was a government agency and top-level administrative department set up in Khanbaliq that supervised Buddhist monks in addition to managing the territory of Tibet. Besides modern-day Tibet Autonomous Region, it also governed a part of Sichuan, Qinghai and Kashmir. It was separate from the other provinces of the Yuan dynasty such as those of former Song dynasty of China, but still under the administrative rule of the Yuan. While no modern equivalents remain, the political functions of the Bureau of Buddhist and Tibetan Affairs might have been analogous to the India Office in London during the British Raj. To emphasize its importance for Hangzhou, the capital of the former Southern Song dynasty, a branch (行) Xuanzheng Yuan was established in that city in 1291.