Digital competence has become increasingly important in modern societies and is today central to the possibility of participating on equal terms as a citizen in a contemporary democracy. Thus, it is now stressed as a crucial learning objective, nationally as well as internationally. One pervasive consequence of the digitalization of society is the facilitation of intrusive online surveillance: when we are online, we leave traces that provide useful information to companies, organizations, and individuals, who can collect, process, use, and share this information. The purpose of this article is to reveal the need for an increased awareness of the surveillance aspect of digitalization in teacher education and schools. The argument is partly based on a questionnaire survey with 560 current and former Swedish student teachers, about online behavior and privacy. The results indicate that Swedish teachers in general need to further their digital competence in order to be able to appropriately aid their pupils in developing digital literacy. Given that Swedish student teachers can be expected to possess a comparatively very high level of digital competence, we think it is safe to generalize this point to comprise teachers in many other countries as well. We argue that an awareness of the surveillance aspect of digitalization is crucial to being a cognizant citizen in a democratic society, and that it should therefore constitute a natural part of education for digital competence. Keywords: education for digital competence, digital competence, education for digital literacy, digital literacy, digitalization, online surveillance, soft surveillance, surveillance, surveillance culture, privacy