The Future Circular Collider (FCC) is a post-LHC project in a new 100-km tunnel under consideration at CERN, that would provide hadron and e+e- collisions at much higher energies and luminosities than reached ever before. The physics program accessible in e+e- collisions at such an FCC-ee facility will be reviewed. The FCC-ee aims at collecting multi-ab−1 integrated luminosities in e+e- at sqrt(s) = 90, 160, 240, and 350 GeV, yielding 1012 Z bosons, 108 W+W− pairs, 106 Higgs bosons and 4⋅10^5 top-quark pairs per year. Such huge and "clean" data samples combined with a O(100keV) c.m. energy uncertainty will allow for Standard Model measurements in the electroweak, Higgs, and top-quark sectors with unparalleled precision, and thereby searches for new physics in regions not probed so far. Among many research goals, the FCC-ee will be able to (i) indirectly discover new particles coupling to the Higgs and electroweak bosons up to scales Λ ≈ 7 and 100 TeV respectively; (ii) measure the Higgs Yukawa couplings to the 1st generation fermions (u,d,s, e+/-), and (iii) achieve the best potential in direct collider searches for dark matter and for sterile neutrinos with masses below 60 GeV.