  We review highlights from string theory, black hole physics and doubly
special relativity and some "thought" experiments which were suggested to probe
the shortest distance and/or the maximum momentum at the Planck scale. The
models which are designed to implement the minimal length scale and/or the
maximum momentum in different physical systems are analysed entered the
literature as the Generalized Uncertainty Principle (GUP). We compare between
them. The existence of a minimal length and a maximum momentum accuracy is
preferred by various physical observations. Furthermore, assuming modified
dispersion relation allows for a wide range of applications in estimating, for
example, the inflationary parameters, Lorentz invariance violation, black hole
thermodynamics, Saleker-Wigner inequalities, entropic nature of the
gravitational laws, Friedmann equations, minimal time measurement and
thermodynamics of the high-energy collisions. One of the higher-order GUP
approaches gives predictions for the minimal length uncertainty. Another one
predicts a maximum momentum and a minimal length uncertainty, simultaneously.
An extensive comparison between the different GUP approaches is summarized. We
also discuss the GUP impacts on the equivalence principles including the
universality of the gravitational redshift and the free fall and law of
reciprocal action and on the kinetic energy of composite system. The concern
about the compatibility with the equivalence principles, the universality of
gravitational redshift and the free fall and law of reciprocal action should be
addressed. We conclude that the value of the GUP parameters remain a puzzle to
be verified.
