The problem of Dark Matter is one of the most longstanding problems in cosmology and particle physics, and surely one of the most interesting. A number of astrophysical and cosmological evidences point to the existence of a large amount of non luminous matter, whose nature is still a mistery. The simplest possible solution to this puzzle is in terms of new neutral and stable particles with weak-scale mass and couplings (named WIMPs for weakly interacting massive particles), which would not only reproduce naturally the observed cosmological Dark Matter abundance, but also fit very well in many well-motivated theories for physics beyond the Standard Model. In this thesis I discuss the present status of WIMP searches and of our understanding of the theoretical issues involved in them, with a particular emphasis to searches at the Large Hadron Collider.