  This article reviews the role of hidden symmetries of dynamics in the study
of physical systems, from the basic concepts of symmetries in phase space to
the forefront of current research. Such symmetries emerge naturally in the
description of physical systems as varied as non-relativistic, relativistic,
with or without gravity, classical or quantum, and are related to the existence
of conserved quantities of the dynamics and integrability. In recent years
their study has grown intensively, due to the discovery of non-trivial examples
that apply to different types of theories and different numbers of dimensions.
Applications encompass the study of integrable systems such as spinning tops,
the Calogero model, systems described by the Lax equation, the physics of
higher dimensional black holes, the Dirac equation, supergravity with and
without fluxes, providing a tool to probe the dynamics of non-linear systems.
