The EU’s actions show how the exercise of extraterritorial jurisdiction by one actor (the EU) based on local approaches to human rights standards and specific values (privacy and the protection of personal data) could lead to the convergence of values and laws on the global stage. Likely directions are decreasing territorialism or broad interpretations of ‘territory’, increasing elevation of fundamental rights to the disadvantage of certain competing interests, and the EU acting to set a high global data protection norm, enabled by the fundamental right to data protection conditioning its exercise of extraterritorial jurisdiction. Convergence could be resisted. If, however, the EU’s reach were strong enough to avoid or counter resistance, this would ultimately lead to fewer conflicts in jurisdiction as global standards would converge and, even in the EU–US data privacy law interface, commonalities and shared approaches to rights protection would emerge.