  By using the 3+1 point of view and parametrized Minkowski theories we develop
the theory of {\it non-inertial} frames in Minkowski space-time. The transition
from a non-inertial frame to another one is a gauge transformation connecting
the respective notions of instantaneous 3-space (clock synchronization
convention) and of the 3-coordinates inside them. As a particular case we get
the extension of the inertial rest-frame instant form of dynamics to the
non-inertial rest-frame one. We show that every isolated system can be
described as an external decoupled non-covariant canonical center of mass
(described by frozen Jacobi data) carrying a pole-dipole structure: the
invariant mass and an effective spin. Moreover we identify the constraints
eliminating the internal 3-center of mass inside the instantaneous 3-spaces.
  In the case of the isolated system of positive-energy scalar particles with
Grassmann-valued electric charges plus the electro-magnetic field we obtain
both Maxwell equations and their Hamiltonian description in non-inertial
frames. Then by means of a non-covariant decomposition we define the
non-inertial radiation gauge and we find the form of the non-covariant Coulomb
potential. We identify the coordinate-dependent relativistic inertial
potentials and we show that they have the correct Newtonian limit.
  Then we study properties of Maxwell equations in non-inertial frames like the
wrap-up effect and the Faraday rotation in astrophysics. Also the 3+1
description without coordinate-singularities of the rotating disk and the
Sagnac effect are given, with added comments on pulsar magnetosphere and on a
relativistic extension of the Earth-fixed coordinate system.
