  CP violation in the Higgs couplings to fermions is an intriguing, but not yet
extensively explored possibility. We use inclusive and differential LHC Higgs
boson measurements to fit the CP structure of the Higgs Yukawa couplings.
Starting with simple effective models featuring CP violation in a single
Higgs-fermion coupling, we probe well-motivated models with up to nine free
parameters. We also investigate the complementarity of LHC constraints with the
electron electric dipole moment bound, taking into account the possibility of a
modified electron Yukawa coupling, and assess to which extent CP violation in
the Higgs-fermion couplings can contribute to the observed baryon asymmetry of
the universe. Even after including the recent analysis of angular correlations
in $H\to\tau^+\tau^-$ decays, we find that a complex tau Yukawa coupling alone
may be able to account for the observed baryon asymmetry, but with large
uncertainties in the baryogenesis calculation. A combination of complex top and
bottom quark Yukawa couplings yields a result four times larger than the sum of
their separate contributions, but remains insufficient to account for the
observed baryon asymmetry.
