  Gauge invariance was discovered in the development of classical
electromagnetism and was required when the latter was formulated in terms of
the scalar and vector potentials. It is now considered to be a fundamental
principle of nature, stating that different forms of these potentials yield the
same physical description: they describe the same electromagnetic field as long
as they are related to each other by gauge transformations. Gauge invariance
can also be included into the quantum description of matter interacting with an
electromagnetic field by assuming that the wave function transforms under a
given local unitary transformation. The result of this procedure is a quantum
theory describing the coupling of electrons, nuclei and photons. Therefore, it
is a very important concept: it is used in almost every fields of physics and
it has been generalized to describe electroweak and strong interactions in the
standard model of particles. A review of quantum mechanical gauge invariance
and general unitary transformations is presented for atoms and molecules in
interaction with intense short laser pulses, spanning the perturbative to
highly nonlinear nonperturbative interaction regimes. Various unitary
transformations for single spinless particle Time Dependent Schr\"odinger
Equations, TDSE, are shown to correspond to different time-dependent
Hamiltonians and wave functions. Accuracy of approximation methods involved in
solutions of TDSE's such as perturbation theory and popular numerical methods
depend on gauge or representation choices which can be more convenient due to
faster convergence criteria. We focus on three main representations: length and
velocity gauges, in addition to the acceleration form which is not a gauge, to
describe perturbative and nonperturbative radiative interactions. Numerical
schemes for solving TDSE's in different representations are also discussed.
