With its promise to transform how we live, work, and think, Big Data has captured the imaginations of governments, businesses, and academia. However, the grand claims of Big Data advocates have been accompanied with concerns about potential detrimental implications for civil rights and liberties, leading to a climate of clash and mutual distrust between different stakeholders. Throughout the years, the interdisciplinary field of technology assessment (TA) has gained considerable experience in studying socio-technical controversies and as such is exceptionally well equipped to assess the premises and implications of Big Data practices. However, the relationship between Big Data as a socio-technical phenomenon and TA as a discipline assessing such phenomena is a peculiar one: Big Data may be the first topic TA deals with that is not only an object of inquiry, but also a major competitor, rivaling TA in several of its core functions, including the assessment of public views and visions, means and methods for exploring the future, and the provision of actionable knowledge and advice for political decision-making. Our paper explores this dual relationship between Big Data and TA before concluding with some considerations on how TA might contribute to more responsible data-based research and innovation.