It is well known that generic two-Higgs-doublet models (2HDMs) suffer from
potentially large Higgs-mediated flavor-changing neutral current (FCNC) problem,
unless additional symmetries are imposed on the Higgs fields thereby respecting the
Natural Flavor Conservation Criterion (NFC) by Glashow and Weinberg. A common
way to respect the NFC is to impose Z_2 symmetry which is softly broken by a dim-2
operator. Another new way is to introduce local U(1)_H Higgs flavor symmetry that
distinguishes one Higgs doublet from the other. In this talk, we discuss the
phenomenology in 2HDMs with not only the Z_2 but also U(1)_H symmetry. We also
study dark matter (DM) physics in the Type-I inert 2HDM with local U(1)_H Higgs
gauge symmetry. The lightest neutral scalar component H of the U(1)_H-charged
Higgs doublet, which does not have Yukawa couplings with the Standard-Model (SM)
fermions, is stable because of the remnant discrete symmetry, and it interacts with the
SM particles through the U(1)_H gauge boson (Z_H) exchange as well as the SM
boson exchange. We investigate the constraints on DM: thermal relic density, and
direct/indirect detections. The additional U(1)_H gauge interaction plays a crucial
role in reducing the DM thermal relic density. The most important result within the
inert DM model with local U(1)_H symmetry is that ~ O(10) GeV dark matter scenario,
which is strongly disfavored in the usual Inert Doublet Model (IDM) with Z_2
symmetry, is revived in our model because of newly open channels, H H -> Z_H Z_H ,
Z_H Z. Exotic Higgs decays, h -> Z_H Z_H, Z Z_H, would be distinctive signatures of
the inert 2HDM with local U(1)_H symmetry. I also give a comment on the type-II
2HDM inspired by E_6 GUT, and introduce some results on DM physics.