The Standard Model (SM) of particle physics summarize our current understanding of nature to its shortest distances. After an impressive forty-year search, the discovery of the Higgs boson by the ATLAS and CMS collaborations at the Large Hadron Collider (LHC) at CERN provided the last experimental proof of such an incredible theoretical construction. Moreover, since the Higgs boson is responsible for making electromagnetism and the weak interaction behave differently, the finding of the long-sought particle also offers us the unique possibility to start testing whether this is a consequence of some underlying dynamics. In particular, this means that we could be closer than ever to understand some extremely important unsolved puzzles in particle physics, like why is the gravitational interaction so much weaker than the electroweak force (the hierarchy problem) or why are the different fermion masses so different (the flavour puzzle). A very appealing theoretical framework trying to provide a solution for some of these questions is given by composite Higgs models. The idea is that, contrary to the SM prediction, the Higgs boson might not be an elementary particle but rather be made of some fermion constituents bounded together by a new strong interaction (similarly to what happens for instance to pions in quantum chromodynamics). At very short distances the new fermion constituents of the Higgs would start to be probed, protecting its mass against higher energy quantum corrections and in particular against those of gravity. Moreover, the large hierarchies existing between the different SM fermion masses could also be explained if one assumes that such fields are a linear superposition of elementary and composite degrees of freedom (dubbed 'partial compositeness'). Indeed, since these masses are generated through the interaction with the Higgs and therefore via the new strong dynamics, they can be reframed naturally in terms of different degrees of compositeness. An interesting consequence of this setup is that the Higgs mass and its self-interactions are generated by quantum effects involving fermions with a sizable degree of compositeness, providing a dynamical origin for these parameters. This establishes a beautiful interplay between the two aforementioned problems. Since the top quark is the heaviest fermion in the SM, it is typically the particle with the largest degree of compositeness and therefore the one providing the leading contribution to these quantum effects. Also, this generally requires that some of the bound states from the new strong sector mixing with the top quark become lighter than the confinement scale (where the different constituents in the strong sector can no longer be found separately), making a priori these new resonances accessible at the LHC. However, current experimental searches performed by both ATLAS and CMS on these 'top partners' have put stringent bounds on how light these particles can be, creating some tension in these models. One interesting possibility presented in the project and that has been explored during this action is that of leptons being partially composite and playing a non-negligible role in the generation of the Higgs mass. Since they and their partners from the new strong sector are colorless (i.e. do not interact via the SM strong interaction), they are produced far less frequently at the LHC. Therefore, current experimental searches impose much milder bounds. Interestingly enough, when one tries to explain the smallness of the neutrino masses in these setups, the requirement of minimality makes the degree of compositeness of the different charged leptons a direct consequence of the tiny neutrino masses, extending non-trivially to the lepton sector the very same logic functioning in the quark one. Since models of composite Higgs are certainly among the best motivated and attractive scenarios trying to address several of the most important puzzles in our current understanding of nature, exploring in detail such possibility becomes very important in order to assess the feasibility of such models. Addressing the different theoretical challenges coming along with these scenarios featuring partially composite leptons, as well as studying the different experimental consequences at colliders, flavor experiments and in Dark Matter searches constitutes the main goals of this project. One of the most exciting conclusions of this research is that the absence of 'top partners' at the LHC can be traded by the observation of B-mesons decaying differently to the various SM charged leptons as hinted by current LHCb data, while meeting all present experimental constraints. Moreover, these scenarios can also accommodate an electrically neutral scalar particle accounting for part or the totality of the Dark Matter relic abundance.
