  The order parameter and its variations in space and time in many different
states in condensed matter physics at low temperatures are described by the
complex function $\Psi({\bf r}, t)$. These states include superfluids,
superconductors, and a subclass of antiferromagnets and charge-density waves.
The collective fluctuations in the ordered state may then be categorized as
oscillations of phase and amplitude of $\Psi({\bf r}, t)$. The phase
oscillations are the {\it Goldstone} modes of the broken continuous symmetry.
The amplitude modes, even at long wavelengths, are well defined and decoupled
from the phase oscillations only near particle-hole symmetry, where the
equations of motion have an effective Lorentz symmetry as in particle physics,
and if there are no significant avenues for decay into other excitations. They
bear close correspondence with the so-called {\it Higgs} modes in particle
physics, whose prediction and discovery is very important for the standard
model of particle physics. In this review, we discuss the theory and the
possible observation of the amplitude or Higgs modes in condensed matter
physics -- in superconductors, cold-atoms in periodic lattices, and in uniaxial
antiferromagnets. We discuss the necessity for at least approximate
particle-hole symmetry as well as the special conditions required to couple to
such modes because, being scalars, they do not couple linearly to the usual
condensed matter probes.
