  The governing equations of Maxwell electrodynamics in multi-dimensional
spaces are derived from the variational principle of least action which is
applied to the action function of the electromagnetic field. The Hamiltonian
approach for the electromagnetic field in multi-dimensional pseudo-Euclidean
(flat) spaces has also been developed and investigated. Based on the two
arising first-class constraints we have generalized to multi-dimensional spaces
a number of different gauges known for the three-dimensional electromagnetic
field. For multi-dimensional spaces of non-zero curvature the governing
equations for the multi-dimensional electromagnetic field are written in
manifestly covariant form. Multi-dimensional Einstein's equations of metric
gravity in the presence of electromagnetic field have been re-written in the
true tensor form. Methods of scalar electrodynamics are applied to analyze
Maxwell equations in the two- and one-dimensional spaces.
