Recent initiatives, such as the United Nations declaring 2015 as the International Year of Soils and the French "4 per 1000" initiative call attention on soils and on the importance of maintaining and increasing soil organic matter stocks for soil fertility and food security, and for climate change adaptation and mitigation. We stress that soil organic carbon storage (i.e. an increase of soil organic carbon stocks) should be clearly differentiated from soil organic carbon sequestration, as the latter assumes a net removal of atmospheric CO2. Implementing management options that allow increasing soil organic carbon stocks at the local scale raises several questions, which are discussed in this article: how can we increase SOC stocks, at which rate and for how long; where do we prioritize SOC storage; how do we estimate the potential gain in C and which agricultural practices should we implement? We show that knowledge and tools are available to answer many of these questions, while further research remains necessary for others. A range of agricultural practices would require a re-assessment of their potential to store C and a better understanding of the underlying processes, such as no tillage and conservation agriculture, irrigation, practices increasing below ground inputs, organic amendments, and N fertilization. The vision emerging from the literature, showing the prominent role of soil microorganisms in the stabilization of soil organic matter, draw the attention to more exploratory potential levers, through changes in microbial physiology or soil biodiversity induced by agricultural practices, that require in-depth research.