Understanding of the neutron star surface is essential to interpret observations of thermal emission. Beyond the determination of the surface temperature, radii are being measured, and one can potentially also measure the surface redshift, permitting deduction of the neutron star mass. For neutron stars with strong magnetic magnetic fields, even the best investigated hydrogen atmosphere models may need some revision (e.g., various hydrogen molecules and molecular ions should be included). Moreover, since some observation indicate the presence of heavier elements (e.g., He and O), the corresponding atmosphere models should be developed for proper interpretation of observations. The possibility of a magnetic solid surface and its emission properties are also to be fully investigated.

Depending on the early history of the star and possible fall-back and/or accretion after the supernova explosion, the chemical composition of matter below the surface may not be "fully catalyzed." Besides its chemical composition, the transport properties of the crust depend on its crystal or amorphous state, and can also be affected significantly by strong magnetic fields. Magnetars are prime examples in which strong magnetic fields are certainly present: their thermal, and mechanical, responses to outbursts can also potentially tell us about the structure of the crust.

The densest layers of the crust can be in a "pasta" phase in which nuclear clusters of various shapes and dimensionality replace spherical nuclei; the structure and properties of this region are only beginning to be understood. The basic transport, both thermal and electrical, characteristics of this region have to be established as they are also potentialy important in absorption of neutrinos during the proto-neutron star phase. The ability, or inability, of the crust to pin the vortices of a neutron superfluid can have significant impact on the rotational evolution of the star and its glitches.

Studies of the evolution of the crust under accretion, both in the post supernova phase during its formation and in old stars accreting from a companion in a binary system, can shed light on much of its properties.