  Quantum eigenstates undergoing cyclic changes acquire a phase factor of
geometric origin. This phase, known as the Berry phase, or the geometric phase,
has found applications in a wide range of disciplines throughout physics,
including atomic and molecular physics, condensed matter physics, optics, and
classical dynamics. In this article, the basic theory of the geometric phase is
presented along with a number of representative applications.
  The article begins with an account of the geometric phase for cyclic
adiabatic evolutions. An elementary derivation is given along with a worked
example for two-state systems. The implications of time-reversal are explained,
as is the fundamental connection between the geometric phase and energy level
degeneracies. We also discuss methods of experimental observation. A brief
account is given of geometric magnetism; this is a Lorenz-like force of
geometric origin which appears in the dynamics of slow systems coupled to fast
ones.
  A number of theoretical developments of the geometric phase are presented.
These include an informal discussion of fibre bundles, and generalizations of
the geometric phase to degenerate eigenstates (the nonabelian case) and to
nonadiabatic evolution. There follows an account of applications.
Manifestations in classical physics include the Hannay angle and kinematic
geometric phases. Applications in optics concern polarization dynamics,
including the theory and observation of Pancharatnam's phase. Applications in
molecular physics include the molecular Aharonov-Bohm effect and nuclear
magnetic resonance studies. In condensed matter physics, we discuss the role of
the geometric phase in the theory of the quantum Hall effect.
