Neutron shell-structure and the resulting possible deformation in the neighborhood of neutrondrip-line nuclei are systematically discussed, based on both bound and resonant neutron oneparticle energies obtained from spherical and deformed Woods-Saxon potentials. Due to the unique behavior of weakly-bound and resonant neutron one-particle levels with smaller orbital angularmomenta ℓ, a systematic change of the shell structure and thereby the change of neutron magicnumbers are pointed out, compared with those of stable nuclei expected from the conventional j-j shell-model. For spherical shape with the operator of the spin-orbit potential conventionally used, the ℓ j levels belonging to a given oscillator major shell with parallel spin-and orbital-angularmomenta tend to gather together in the energetically lower half of the major shell, while those levels with anti-parallel spin-and orbital-angular-momenta gather in the upper half. The tendency leads to a unique shell structure and possible deformation when neutrons start to occupy the orbits in the lower half of the major shell. Among others, the neutron magic-number N=28 disappears and N=50 may disappear, while the magic number N=82 may presumably survive due to the large ℓ = 5 spin-orbit splitting for the 1h 11/2 orbit. On the other hand, an appreciable amount of energy gap may appear at N=16 and 40 for spherical shape, while neutron-drip-line nuclei in the region of neutron number above N=20, 40 and 82, namely N ≈ 21-28, N ≈ 41-54, and N ≈ 83-90, may be quadrupole-deformed though the possible deformation depends also on the proton number of respective nuclei.