Organic conservation agriculture can potentially contribute to meeting many urgent technical and environmental issues related to agricultural soil degradation; however, its uptake has been so far quite limited. After having shed some light on the similarities and differences between (organic) conservation agriculture and the emerging concept of (organic) regenerative agriculture, the authors tackle the problem of weed management in such systems, which is one of the major technical barriers against adoption. The weed management challenges posed by organic conservation/regenerative systems can best be tackled through cropping system diversification. In this respect, cover crops/dead mulches and intercrops/living mulches appear as interesting practices to improve weed suppression. They highlight the key theoretical and practical key issues linked to the efficiency and optimisation of such practices, and provide some examples of success stories. Finally, they present some opportunities, challenges and future perspectives of weed management in organic conservation and regenerative systems.