After the Higgs boson discovery, the precision measurements and searches for new phenomena in the Higgs sector are among the most important goals in particle physics. Experiments at the Future Circular Colliders (FCC) are ideal to study these questions. Electron-positron collisions up to an energy of 350 GeV (FCC-ee) provide the ultimate precision with studies of Higgs boson couplings, mass, total width and CP parameters, as well as searches for exotic and invisible decays. The feasibility of observation of the s-channel production e+e- → H(125) is reviewed. We conclude by noting the remarkable complementarity of the FCC-ee and FCC-hh colliders, which in combination offer the best possible overall study of the Higgs boson properties.