Measurements of Higgs production and decays have revealed that most of the electroweak symmetry breaking is due to the 125 GeV Higgs field. Similarly, we know that the Higgs is at least partially responsible for giving mass to the top and bottom quarks, and tau lepton. Much less is known about the origin of mass for the first two generations, as well as about possible flavor violating Higgs couplings. In this talk, I discuss the present status of the measurements of these challenging Higgs couplings, and the New Physics implications of a possible future observation of Higgs flavor violating couplings.