Core Legal Answer & Context: Asylum is the protection granted by a state to a person who has fled their home country due to persecution or a well-founded fear of persecution.

1. Territorial Asylum: This is the most common form, where a state grants asylum to an individual within its own territory. The asylum seeker physically enters the territory of the asylum-granting state and is protected under its domestic laws. Granting territorial asylum is a sovereign right of states, though it is often guided by international refugee law.
2. Extraterritorial Asylum (or Diplomatic Asylum): This is granted by a state within its diplomatic missions (embassies, consulates) or warships/military aircraft located in the territory of another state. It is an exceptional and controversial practice because it involves granting protection outside the asylum-granting state's own territory, potentially infringing on the territorial sovereignty of the host state. It is typically granted to political offenders.

Relevant Statutes and Sections:

1. Universal Declaration of Human Rights (UDHR), Article 14: States that everyone has the right to seek and to enjoy in other countries asylum from persecution.
2. Convention Relating to the Status of Refugees (1951) and its 1967 Protocol: Govern territorial asylum for refugees.
3. Convention on Diplomatic Asylum (1954) (OAS): A regional convention primarily applicable in Latin America, attempting to regulate diplomatic asylum.

Important Landmark Cases:

1. Asylum Case (Colombia v. Peru) (ICJ, 1950): This case dealt with diplomatic asylum, where Colombia granted asylum to a Peruvian political leader in its embassy. The ICJ ruled that Colombia could not unilaterally determine the nature of the offense and that there was no customary international law right to grant diplomatic asylum in such circumstances, highlighting its contentious nature and limited scope.

Clear Conclusion: Territorial asylum is the widely accepted practice of granting protection within a state's borders. Extraterritorial (diplomatic) asylum, granted within diplomatic premises in another state, is an exceptional and more contentious practice, often limited by specific regional conventions or special circumstances, and is generally not recognized as a right under customary international law.