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Observations of magnetars and some of the high magnetic field pulsars have shown that their thermal luminosity is systematically higher than that of classical radiopulsars, thus confirming the idea that magnetic fields are involved in their X-ray emission. Here we present the results of 2D simulations of the fully-coupled evolution of temperature and magnetic field in neutron stars, including the state-of-the-art kinetic coefficients and, for the first time, the important effect of the Hall term. After gathering and thoroughly re-analysing in a consistent way all the best available data on isolated, thermally emitting neutron stars, we compare our theoretical models to a data sample of 40 sources. We find that our evolutionary models can explain the phenomenological diversity of magnetars, high-B radio-pulsars, and isolated nearby neutron stars by only varying their initial magnetic field, mass and envelope composition. Nearly all sources appear to follow the expectations of the standard theoretical models. Finally, we discuss the expected outburst rates and the evolutionary links between different classes. Our results constitute a major step towards the grand unification of the isolated neutron star zoo.with the very active magnetars, showing intermediate luminosities, and sporadic magnetar-like activity. The radioquiet X-ray Isolated NSs (XINSs) are a class of relatively old, nearby cooling NSs, with the cleanest detected thermal emission and a relatively large magnetic field. Last, Central Compact Objects (CCOs, Gotthelf et al. 2013) represent a handful of puzzling, radio-quiet sources which in some cases combine a very weak external magnetic field with a relatively large luminosity and evidence for anisotropic surface temperature distribution.Although isolated NSs have been divided into all these observational classes for historical reasons, a unifying vision considers them as different manifestations of the same underlying physics (Kaspi 2010). In this context, one of the main theoretical tasks is to explain the varied phenomenology of their X-ray emission. X-ray spectra carry precious information about the surface temperature and the physics driving the cooling of the NSs. The detected X-ray flux, if accompanied by a reliable distance measurement, leads to an estimate of the bolometric luminosity, which can be confronted with cooling models to infer properties of dense matter in the NS interior. In addition, timing properties (period

Context. The presence of magnetic fields in the crust of neutron stars (NSs) causes a non-spherically symmetric temperature distribution. The strong temperature dependence of the magnetic diffusivity and thermal conductivity, together with the heat generated by magnetic dissipation, couple the magnetic and thermal evolution of NSs, which can no longer be formulated as separated onedimensional problems. Aims. We study the mutual influence of thermal and magnetic evolution in a neutron star's crust in axial symmetry. Taking realistic microphysical inputs into account, we find the heat released by Joule effect consistent with the circulation of currents in the crust, and we incorporate its effects in 2D cooling calculations. Methods. We solve the induction equation numerically using a hybrid method (spectral in angles, but a finite-differences scheme in the radial direction), coupled to the thermal diffusion equation. To improve the boundary conditions, we also revisit the envelope stationary solutions updating the well known T b − T s -relations to include the effect of 2D heat transfer calculations and new microphysical inputs.Results. We present the first longterm 2D simulations of the coupled magneto-thermal evolution of neutron stars. This substantially improves previous works in which a very crude approximation in at least one of the parts (thermal or magnetic diffusion) has been adopted. Our results show that the feedback between Joule heating and magnetic diffusion is strong, resulting in a faster dissipation of the stronger fields during the first 10 5 −10 6 years of an NS's life. As a consequence, all neutron stars born with fields over a critical value (>5 × 10 13 G) reach similar field strengths (≈2−3 × 10 13 G) at late times. Irrespective of the initial magnetic field strength, the temperature becomes so low after 10 6 years that the magnetic diffusion timescale becomes longer than the typical ages of radiopulsars, thus apparently resulting in no dissipation of the field in old NS. We also confirm the strong correlation between the magnetic field and the surface temperature of relatively young NSs discussed in preliminary works. The effective temperature of models with strong internal toroidal components are systematically higher than those of models with purely poloidal fields, due to the additional energy reservoir stored in the toroidal field that is gradually released as the field dissipates.