The flows of people, goods and capital, which have considerably increased in recent history, are leading to crises (e.g., migrants, tax evasion, food safety) which reveal the failure to control them. Much less visible, and not yet included in economic measurements, data flows have increased exponentially in the last two decades, with the digitisation of social and economic activities. A new space – Datasphere – is emerging, mostly supported by digital platforms which provide essential services reaching half of the world’s population directly. Their control over data flows raises new challenges to governance, and increasingly conflicts with public administration. In this paper, we consider the need and the difficulty of regulating this emerging space and the different approaches followed on both sides of the Atlantic. We distinguish between three situations. We first consider data at rest, which is from the point of view of the location where data are physically stored. We then consider data in motion, and the issues related to their combination. Finally, we investigate data in action, that is data as vectors of command of legal or illegal activities over territories, with impacts on economy and society as well as security, and raise governance challenges. The notion of ‘Datasphere’ proposes a holistic comprehension of all the ‘information’ existing on earth, originating both in natural and socio-economic systems, which can be captured in digital form, flows through networks, and is stored, processed and transformed by machines. It differs from the ‘Cyberspace’, which is mostly concerned with the networks, the technical instruments (from software and protocols to cables and data centers) together with the social activities it allows, and to what extent they could/should be allowed. The paper suggests one – out of the many possible – approach to this new world. Clearly it would be impossible to delve in depth into all its facets, which are as many as those of the physical world. Rather, it attempts to present how traditional legal notions could be usefully managed to put order in a highly complex environment, avoiding a piecemeal approach that looks only at details.