A Higgs boson with richer flavor interactions arises when the flavor structure encoded in supersymmetric extensions of the standard model is transmitted to the Higgs sector. The flavor-Higgs transmission mechanism can have a radiative or mixing origin, as it is illustrated with several examples, and can produce interesting Higgs signatures that can be probed at future high-energy colliders. Within the minimal SUSY extension of the SM (MSSM), the flavor mediation mechanism is of radiative type, as it is realized through gaugino-sfermion loops, and it transmits the flavor structure of the soft-breaking sector to the Higgs bosons; for this case we evaluate the contributions from general trilinear A-terms to the Lepton Flavor-Violating (LFV) and Flavor-Conserving (LFC) Higgs vertices. On the other hand, as an example of flavor mediation through mixing, we discuss an E-6-inspired multi-Higgs model, supplemented with an abelian flavor symmetry, where LFV as well as LFC Higgs effects, are found to arise, though in this case at tree-level. We find that Tevatron and LHC can provide information on the flavor structure of these models through the detection of the LFV Higgs mode h --> taumu, while NLC can perform high-precision tests of the LFC mode h --> tau(+)tau(-).