  Theoretical and experimental studies of Berry and Pancharatnam phases are
reviewed. Basic elements of differential geometry are presented for
understanding the topological nature of these phases. The basic theory analyzed
by Berry in relation to magnetic monopoles is presented. The theory is
generalized to nonadiabatic processes and to noncyclic Pancharatnam phases.
Different systems are discussed including polarization optics, n-level atomic
systems, neutron interferometry and molecular topological phases.
