In 2012 two LHC experiments have discovered a new particle with a mass around 125 GeV, which might be the scalar Higgs boson of the Standard Model. This particle could be produced in large numbers for precision studies by an e+e- collider operating near the ZH threshold at a beam energy of 120 GeV or, in the s-channel, by a gamma-gamma collider with primary electron beam energies of 80 GeV. In this seminar I will discuss tentative design parameters, novel concepts and accelerator-physics challenges for two circular e+e- Higgs-factory colliders - LEP3 and TLEP - and for a gamma-gamma Higgs-factory collider based on a recirculating SC electron linac - SAPPHiRE. LEP3, installed in the existing 27-km LHC tunnel, and TLEP, in a new 80-km long tunnel, require - in addition to the collider ring - a fast cycling accelerator ring for quasi-continuous top-up injection to compensate the short beam lifetime due to radiative Bhabha scattering at luminosities above 1e34 cm-2s-1. They also need a large momentum aperture to ensure an adequate beam lifetime with regard to beamstrahlung. The 80-km tunnel of TLEP could later accommodate a very high energy LHC ("VHE-LHC"), with a centre-of-mass energy approaching 100 TeV in pp collisions. An important challenge for SAPPHiRE is the layout of the interaction region and the generation of the high-power photon pulses needed for Compton back scattering. These photon pulses, which collide with the electron beams about 1 mm from the interaction point proper, could be produced either by a conventional laser together with an optical stacking cavity or, more intriguingly, by an FEL process, possibly driven by the 80-GeV electron bunches themselves. SAPPHiRE could be reconfigured as "LHeC" (or vice versa), providing a 60-GeV electron beam of higher current in energy-recovery mode to collide with one of the LHC hadron beams. SAPPHiRE could also be based on infrastructures and expertise available at DESY, FNAL, KEK or TJNAF.