Constraints on effective field theories by unitarity and crossing symmetry

In this thesis we discussed the fundamental consistency conditions that Wilson coefficients must respect in order for an EFT to admit a UV completion that is Lorentz invariant, unitarity and crossing symmetric. These consistency conditions take the form of sum rules and positivity constraints for the low-energy scattering amplitudes that are expressed in terms of the Wilson coefficients. We extended significantly the results from earlier literature by treating in full generality particles carrying arbitrary spin and arbitrary representations of internal symmetry groups. We applied these positivity constraints to certain EFTs for physics beyond of the SM, and probe them with data from the current run of the LHC.