Globalization confronts the European Union with many new challenges. One of these concerns the applicability of harmonized EU law to cross-border situations involving third countries. In its recent judgment in Google/CNIL (C-507/17), on the territorial reach of the EU data protection rules and the “right to be forgotten”, the CJEU introduces a new “logic of globalization” which must be distinguished from the traditional “logic of the internal market”. While the latter justifies extraterritoriality in case internal market interests are affected, restraint characterizes the former. The global horizon does not diminish pertinent EU interests and objectives, but their effective implementation is threatened by the absence of the ensured enforcement of EU law and potential countermeasures. In context of globalization, it is international collaboration rather than unilateralism that would enable the EU to protect its interests and those of its citizens more adequately.