  We analyze the constraints on CP-violating, flavor conserving Two Higgs
Doublet Models (2HDMs) implied by measurements of Higgs boson properties at the
Large Hadron Collider (LHC) and by the non-observation of permanent electric
dipole moments (EDMs) of molecules, atoms and the neutron. We find that the LHC
and EDM constraints are largely complementary, with the LHC studies
constraining the mixing between the neutral CP-even states and EDMs probing the
effect of mixing between the CP-even and CP-odd scalars. The presently most
stringent constraints are implied by the non-observation of the ThO molecule
EDM signal. Future improvements in the sensitivity of neutron and diamagnetic
atom EDM searches could yield competitive or even more severe constraints. We
analyze the quantitative impact of hadronic and nuclear theory uncertainties on
the interpretation of the latter systems and conclude that these uncertainties
cloud the impact of projected improvements in the corresponding experimental
sensitivities.
