U.S. relations with many countries are governed by a number of treaties, both multilateral and bilateral. The functions of U.S. consular officers to protect U.S. citizens abroad are listed in a multilateral treaty called the Vienna Convention on Consular Relations of 1963 which defines the framework for consular relations between countries. The U.S. also has bilateral treaties with a number of countries concerning consular matters. You can read about these here: Bilateral Consular Conventions.
Private international law is the body of conventions, model laws, national laws, legal guides, and other documents and instruments that regulate private relationships across national borders. These multi-lateral treaties include: