Clarifying the legal status of data is one of the major issues and challenges the European legislator has to deal and cope with while establishing a legal frame for a European data economy. Albeit at the beginning of the policy-making process the idea of creating a data right (on non-personal data) was on the agenda of the European Commission, this idea now seems to be have been abandoned. Instead, the focus actually lies on the creation of access and management regimes (e.g. Open Data Directive, Proposal for a European Data Governance Act). This approach seems, from a private law perspective, questionable, since the access right is inherently connected to – or more precisely, flowing from – an allocative right (in its most classical form: ownership). This article is a plea for a private law, or more precisely, property law perspective, namely for the creation of an allocative data right. It explores why the access approach from a private law perspective appears to be inconsistent and outlines the contours of such allocative right. In this context, contemporary property law theories are also assessed, since it is primarily property law which shall serve as the doctrinal background and framework for the envisaged allocative data right.