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The study of the couplings of the Higgs boson and of the top quark plays a preeminent role at the LHC, and could unveil the first signs of new physics. I will discuss the interplay of direct and indirect probes of certain classes of top and Higgs couplings. Including constraints from collider observables, precision electroweak tests, flavor physics, and electric dipole moments (EDMs), I will show that indirect probes are competitive, if not dominant, for both the CP-even and CP-odd top and Higgs couplings we considered. I will discuss the role of theoretical uncertainties, associated with hadronic and nuclear matrix elements, and indicate targets to further improve the constraining power of EDM experiments.