Nuclei at both the neutron- and proton-drip lines are studied. In the cluster–core model, the halo structure of all the observed and proposed cases of neutron- or proton-halos is investigated in terms of simple potential energy surfaces calculated as the sum of binding energies, Coulomb repulsion, nuclear proximity attraction and the centrifugal potential for all the possible cluster+core configurations of a nucleus. The clusters of neutrons and protons are taken to be unbound, with additional Coulomb energy added for proton-clusters. The model predictions agree with the available experimental studies but show some differences with the nucleon separation energy hypothesis, particularly for proton-halo nuclei. Of particular interest are the halo structures of ¹¹N and ²⁰Mg. The calculated potential energy surfaces are also useful to identify the new magic numbers and molecular structures in exotic nuclei. In particular, N 6 is a possible new magic number for very neutron-deficient nuclei, but N 2, Z 2 and Z 8 seem to remain magic even for such nuclei, near the drip line.