  Neutron stars are compact objects rotating at high speed, up to a substantial
fraction of the speed of light (up to 20\% for millisecond pulsars) and
possessing ultra-strong electromagnetic fields (close to and sometimes above
the quantum critical field of \numprint{4.4e9}~\SIunits{\tesla}). Moreover, due
to copious $e^\pm$ pair creation within the magnetosphere, the relativistic
plasma surrounding the star is forced into corotation up to the light cylinder
where the corotation speed reaches the speed of light. The neutron star
electromagnetic activity is powered by its rotation which becomes relativistic
in the neighbourhood of this light cylinder. These objects naturally induce
relativistic rotation on macroscopic scales about several thousands of
kilometers, a crucial ingredient to trigger the central engine as observed on
Earth. In this paper, we elucidate some of the salient features of this
corotating plasma subject to efficient particle acceleration and radiation,
emphasizing several problems and limitations concerning current theories of
neutron star magnetospheres. Relativistic rotation in these systems is
indirectly probed by the radiation produced within the magnetosphere. Depending
on the underlying assumptions about particle motion and radiation mechanisms,
different signatures on their light-curves, spectra, pulse profiles and
polarisation angles are expected in their broadband electromagnetic emission.
We show that these measurements put stringent constraints on the way to
describe particle electrodynamics in a rotating neutron star magnetosphere.
