  The Compact Linear Collider (CLIC) is an option for a future e+e- collider
operating at centre-of-mass energies up to 3 TeV, providing sensitivity to a
wide range of new physics phenomena and precision physics measurements at the
energy frontier. This paper is the first comprehensive presentation of the
Higgs physics reach of CLIC operating at three energy stages: sqrt(s) = 350
GeV, 1.4 TeV and 3 TeV. The initial stage of operation allows the study of
Higgs boson production in Higgsstrahlung (e+e- -> ZH) and WW-fusion (e+e- ->
Hnunu), resulting in precise measurements of the production cross sections, the
Higgs total decay width Gamma_H, and model-independent determinations of the
Higgs couplings. Operation at sqrt(s) > 1 TeV provides high-statistics samples
of Higgs bosons produced through WW-fusion, enabling tight constraints on the
Higgs boson couplings. Studies of the rarer processes e+e- -> ttH and e+e- ->
HHnunu allow measurements of the top Yukawa coupling and the Higgs boson
self-coupling. This paper presents detailed studies of the precision achievable
with Higgs measurements at CLIC and describes the interpretation of these
measurements in a global fit.
