The roles of isospin asymmetry in nuclei and neutron stars are investigated using a range of potential and field-theoretical models of nucleonic matter. The parameters of these models are fixed by fitting the properties of homogeneous bulk matter and closed-shell nuclei. We discuss and unravel the causes of correlations among the neutron skin thickness in heavy nuclei, the pressure of beta-equilibrated matter at a density of 0.1fm-3, the derivative of the nuclear symmetry energy at the same density and the radii of moderate mass neutron stars. Constraints on the symmetry properties of nuclear matter from the binding energies of nuclei are examined. The extent to which forthcoming neutron skin measurements will further delimit the symmetry properties is investigated. The impact of symmetry energy constraints for the mass and moment of inertia contained within neutron star crusts and the threshold density for the nucleon direct Urca process, all of which are potentially measurable, is explored. We also comment on the minimum neutron star radius, assuming that only nucleonic matter exists within the star.