Dark Matter (DM)-nucleus collisions in underground experiments occur in deeply non-relativistic (NR) regime. Hence, in the former part of this talk, I present a general approach to study signals in direct detection (DD) based on the formalism of NR interactions. Then, I review how to match a non-complete list of effective operators for fermion DM and quarks fields down to NR DM-nucleus interactions and I provide bounds from DD on the energy scale of such operators. Finally, in the framework of simplified model of DM in which the interactions with Standard Model fermions are mediated by a heavy vector boson, I show how the evolution of the couplings between the scale of the mediator mass and the nuclear energy scale changes the bounds computed without including this important effect, as a consequence of operator mixing which generates new interactions at low energy.
