  The FCC-ee offers powerful opportunities to determine the Higgs boson
parameters, exploiting over $10^6$ ${\rm e^+e^- \to ZH}$ events and almost
$10^5$ ${\rm WW \to H}$ events at centre-of-mass energies around 240 and 365
GeV. This essay spotlights the important measurements of the ZH production
cross section and of the Higgs boson mass. The measurement of the total ZH
cross section is an essential input to the absolute determination of the HZZ
coupling -- a "standard candle" that can be used by all other measurements,
including those made at hadron colliders -- at the per-mil level. A combination
of the measured cross sections at the two different centre-of-mass energies
further provides the first evidence for the trilinear Higgs self-coupling, and
possibly its first observation if the cross-section measurement can be made
accurate enough. The determination of the Higgs boson mass with a precision
significantly better than the Higgs boson width (4.1 MeV in the Standard Model)
is a prerequisite to either constrain or measure the electron Yukawa coupling
via direct ${\rm e^+e^- \to H}$ production at $\sqrt{s} = 125$ GeV. Approaching
the statistical limit of 0.1% and $\mathcal{O}(1)$ MeV on the ZH cross section
and the Higgs boson mass, respectively, sets highly demanding requirements on
accelerator operation (ZH threshold scan, centre-of-mass energy measurement),
detector design (lepton momentum resolution, hadronic final state
reconstruction performance), theoretical calculations, and analysis techniques
(efficiency and purity optimization with modern tools, constrained kinematic
fits, control of systematic uncertainties). These challenges are examined in
turn in this essay.
