What is the right to remain in another country if you cannot live safely in your own because of the political situation there?

A government may grant a foreign national asylum in a country to flee persecution in their own country. The hosting country is also said to be providing sanctuary to that foreigner.

Describing it as a right might be misleading. A government may or may not choose to grant asylum. For the United States, it often depends on just why the person is requesting asylum. It may be murderer, child-molester, or some other form of criminal fleeing justice, after all.