  Off-centred dipole configurations have been suggested to explain different
phenomena in neutron stars, such as natal kicks, irregularities in polarisation
of radio pulsars and properties of X-ray emission from millisecond pulsars.
Here for the first time we model magneto-thermal evolution of neutron stars
with crust-confined magnetic fields and off-centred dipole moments. We find
that the dipole shift decays with time if the initial configuration has no
toroidal magnetic field. The decay timescale is inversely proportional to
magnetic field. The octupole moment decreases much faster than the quadrupole.
Alternatively, if the initial condition includes strong dipolar toroidal
magnetic field, the external poloidal magnetic field evolves from centred
dipole to off-centred dipole. The surface thermal maps are very different for
configurations with weak $B = 10^{13}$ G and strong $B = 10^{14}$ G magnetic
fields. In the former case, the magnetic equator is cold while in the latter
case it is hot. We model lightcurves and spectra of our magneto-thermal
configurations. We found that in the case of cold equator, the pulsed fraction
is small (below a few percent in most cases) and spectra are well described
with a single blackbody. Under the same conditions models with stronger
magnetic fields produce lightcurves with pulsed fraction of tens of percent.
Their spectra are significantly better described with two blackbodies. Overall,
the magnetic field strength has a more significant effect on bulk thermal
emission of neutron stars than does the field geometry.
