The uniqueness of auxetic beams, in comparison to conventional ones, is explored in this chapter in terms of axial deformation of prismatic bars, cantilever bending of beams, and torsion of rods. In axial deformation of prismatic bars, it is herein shown that the change in density is more drastic in auxetic solids than in conventional ones. The cantilever bending is herein considered for circular cross sections, as well as rectangular cross sections of various aspect ratios. Emphasis is given to the loci of conventional, moderately auxetic and highly auxetic regions in the analyses. For the special case of twisting of a beam in which one cross section remains plane, there is no diminution of the beam length when the Poisson’s ratio of the beam material is −1.