Algorithmic decision systems are already used in many everyday tools and services on the Internet. They also play an increasing role in many situations in which people’s lives and rights are strongly affected, such as job and loans applications, but also medical diagnosis and therapeutic choices, or legal advice and court decisions. This evolution gives rise to a whole range of questions. In this paper, we argue that certification and explanation are two complementary means of strengthening the European legal framework and enhancing trust in algorithmic decision systems. The former can be seen as the delegation of the task of checking certain criteria to an authority, while the latter allows the stakeholders themselves (for example, developers, users and decision-subjects) to understand the results or the logic of the system. We explore potential legal requirements of accountability in this sense and their effective implementation. These two aspects are tackled from the perspective of the European and French legal frameworks. We focus on two particularly sensitive application domains, namely the medical and legal sectors.