The discovery of a new bosonic resonance, whose interactions are consistent with those of the Standard Model Higgs boson, have been reported by the ATLAS and CMS collaborations. In this thesis we describe the theoretical reasons to consider the Standard Model Higgs sector as an odd and unsatisfying setup to realize Electroweak Symmetry Breaking. We introduce the two broad conceptual frameworks which can be used to extend the Standard Model in order to realize natural Electroweak Symmetry Breaking. These are weak-scale strong dynamics and weak-scale Supersymmetry. We discuss, for both cases, specific model-building and phenomenological aspects related to the implementation of these ideas in extensions of the Standard Mode