It is now widely believed that our Universe experienced
inflationary expansion in the early stage when fluctuations
that seeded the observed large-scale structure were
generated quantum mechanically. During this epoch
the Higgs field in the standard model also acquired
quantum fluctuations. We discuss its cosmological
consequences. In the most dramatic case, the Higgs
field would exhibit a run-away behavior to cause
instability of the Universe. Even if this was not the
case, long-wave fluctuations produce Higgs condensation
in general, which may generate too large curvature
fluctuations in some types of inflation models.
Finally we also discuss the possibility that the Higgs
field itself induced inflation in the early Universe in the
context of the generalized G inflation.