  When a neutron star forms, after the collapse of its progenitor, a strong
magnetic field survives in its interior. This magnetic topology is usually
assumed to be well approximated by a dipole located right at the centre of the
star. However, there is no particular reason why this dipole should be attached
to this very special point. A slight shift from the stellar centre could have
strong implications for the surrounding electromagnetic field configuration
leading to clear observational signatures. We study the effect of the most
general off-centred dipole anchored in the neutron star interior. Exact
analytical solutions are given in vacuum outside the star to any order of
accuracy in the small parameter $\epsilon = d/R$, where $d$ is the displacement
of the dipole from the stellar centre and $R$ the neutron star radius. As a
simple diagnostic of this decentred dipole, the spin-down luminosity and the
torque exerted on its crust are computed to the lowest leading order
in~$\epsilon$. Results are compared to earlier works and a discussion on
repercussions on pulsar braking index and multi-wavelength light curves is
proposed.
