  Whenever a quantum system undergoes a cycle governed by a slow change of
parameters, it acquires a phase factor: the geometric phase. Its most common
formulations are known as the Aharonov-Bohm, Pancharatnam and Berry phases, but
both prior and later manifestations exist. Though traditionally attributed to
the foundations of quantum mechanics, the geometric phase has been generalized
and became increasingly influential in many areas from condensed-matter physics
and optics to high energy and particle physics and from fluid mechanics to
gravity and cosmology. Interestingly, the geometric phase also offers unique
opportunities for quantum information and computation. In this Review we first
introduce the Aharonov-Bohm effect as an important realization of the geometric
phase. Then we discuss in detail the broader meaning, consequences and
realizations of the geometric phase emphasizing the most important mathematical
methods and experimental techniques used in the study of geometric phase, in
particular those related to recent works in optics and condensed-matter
physics.
