Аннотация:The search for a Dark Matter particle is the new grail and hard-sought nirvanaof the particle physics community. From the theoretical side, it is the main challenge toprovide a consistent and model-independent tool for comparing the bounds and reach ofthe diverse experiments. We propose a first complete classification of minimal consistentDark Matter models, abbreviated as MCDMs, that are defined by one Dark Matter weakmultiplet with up to one mediator multiplet. This classification provides the missing linkbetween experiments and top-down models. Consistency is achieved by imposing renormalisability and invariance under the full Standard Model symmetries. We apply thisparadigm to the fermionic Dark Matter case. We also reconsider the one-loop contributions to direct detection, including the relevant effect of (small) mass splits in the Darkmultiplet. Our work highlights the presence of unexplored viable models, and paves theway for the ultimate systematic hunt for the Dark Matter particle.