  In the standard model (SM), the coupling of the Higgs boson to electrons is
real and very small, proportional to the electron mass. New physics could
significantly modify both real and imaginary parts of this coupling. We discuss
experiments which are sensitive to the Higgs-electron coupling and derive the
current bounds on new physics contributing to this coupling. The strongest
constraint follows from the ACME bound on the electron electric dipole moment
(EDM). We calculate the full analytic two-loop result for the electron EDM and
show that it bounds the imaginary part of the Higgs-electron coupling to be
less than 1.7 x 10^-2 times the SM electron Yukawa coupling. Deviations of the
real part are much less constrained. We discuss bounds from Higgs decays,
resonant Higgs production at electron colliders, Higgs mediated B -> e^+ e^-
decays, and the anomalous magnetic moment of the electron. Currently, the
strongest constraint comes from h -> e^+ e^- at the LHC, bounding the coupling
to be less than ~600 times the SM Yukawa coupling. Important improvements can
be expected from future EDM measurements as well as from resonant Higgs
production at a next-generation high-luminosity e^+ e^- collider.
