  Recent progress on nonlinear properties of parity-time ($\cal PT$-) symmetric
systems is comprehensively reviewed in this article. $\cal PT$ symmetry started
out in non-Hermitian quantum mechanics, where complex potentials obeying $\cal
PT$ symmetry could exhibit all-real spectra. This concept later spread out to
optics, Bose-Einstein condensates, electronic circuits, and many other physical
fields, where a judicious balancing of gain and loss constitutes a $\cal
PT$-symmetric system. The natural inclusion of nonlinearity into these $\cal
PT$ systems then gave rise to a wide array of new phenomena which have no
counterparts in traditional dissipative systems. Examples include the existence
of continuous families of nonlinear modes and integrals of motion,
stabilization of nonlinear modes above $\cal PT$-symmetry phase transition,
symmetry breaking of nonlinear modes, distinctive soliton dynamics, and many
others. In this article, nonlinear $\cal PT$-symmetric systems arising from
various physical disciplines are presented; nonlinear properties of these
systems are thoroughly elucidated; and relevant experimental results are
described. In addition, emerging applications of $\cal PT$ symmetry are pointed
out.
