Toward High Precision Higgs-Boson Measurements at the International Linear $e^+e^-$ Collider

128 pages, lots of figures. One subsection added and other minor modifications - Electronic Conference Proceedings (eConf)This report reviews the properties of Higgs bosons in the Standard Model (SM) and its various extensions. We give an extensive overview about the potential of the ILC operated at centre-of-mass energies up to 1 TeV (including the gamma gamma option) for the determination of the Higgs boson properties. This comprises the measurement of the Higgs boson mass, its couplings to SM fermions and gauge bosons, and the determination of the spin and the CP quantum numbers of the Higgs. The extensions of the SM that are analyzed in more detail are heavy SM-like Higgs bosons, heavy Higgs bosons in the framework of Supersymmetry (SUSY) and further exotic scenarios. We review recent theoretical developments in the field of Higgs boson physics. The important question what the ILC can contribute to Higgs boson physics after the LHC, the LHC/ILC interplay and synergy is discussed. The impact of Higgs boson physics on cosmology in several SUSY frameworks is analyzed. The impact of the accelerator and dector performance on the precision of measurements are discussed in detail. We propose a strategy to optimize future analyses. Open questions arising for the various topics are listed, further topics of study and corresponding roadmaps are suggested