  The accelerated expansion of the early universe is an integral part of modern
cosmology and dynamically realized by the mechanism of inflation. The simplest
theoretical description of the inflationary paradigm is based on the assumption
of an additional propagating scalar degree of freedom which drives inflation -
the inflaton. In most models of inflation the fundamental nature of the
inflaton remains unexplained. In the model of Higgs inflation, the inflaton is
identified with the Standard Model Higgs boson and connects cosmology with
elementary particle physics. A characteristic feature of this model is a
non-minimal coupling of the Higgs boson to gravity. I review and discuss
several phenomenological and fundamental aspects of this model, including the
impact of quantum corrections and the renormalization group, the derivation of
initial conditions for Higgs inflation in a quantum cosmological framework and
the classical and quantum equivalence of different field parametrizations.
