We explore the atomic and electronic structures of single-crystalline aluminum nitride nanowires (A1NNWs) and thick-walled aluminum nitride nanotubes (A1NNTs) with the diameters ranging from 0.7 to 2.2 nm by using first-principles calculations and molecular dynamics simulations based on density functional theory (DFT). We find that the preferable lateral facets of A1NNW sand thick-walled A1NNTs are {10 (1) over bar0} surfaces, giving rise to hexagonal cross sections. Quite different from the cylindrical network of hexagons revealed in single-walled A1NNTs, the wall of thick-walled A1NNTs displays a wurtzite structure. The strain energies per atom in A1NNWs are proportional to the inverse of the wire diameter, whereas those in thick-walled A1NNTs are independent of tube diameter but proportional to the inverse of the wall thickness. Thick-walled A1NNTs are energetically comparable to A1NNWs of similar diameter, and both of them are energetically more favorable than single-walled AlNNTs. Both A1NNWs and A1NNTs are wide band gap semiconductors accompanied with surface states located in the band gap of bulk wurtzite A1N.