Greenland is part of the Kingdom of Denmark. However, the local population has been enabled to develop and run their own political system. The monarch of Denmark does have executive power, but she mostly serves in a ceremonial and representative capacity.
Greenland offered a home to Paleo-Eskimo tribes as far back as 2500 B.C.E., but it became settled by Norwegians and Icelanders by 986 C.E. In the 14th Century Norway became a part of Denmark. The Treaty of Kiel gave Denmark final control of Greenland in 1814, but Norway claimed the eastern section of the country. This claim was successfully disputed in 1933, and Denmark has had control of Greenland ever since. The country was granted home rule by Denmark in 1979. Interestingly, the U.S. has shown a high level of interest in Greenland, including once offering to buy the nation for $100 million USD.