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Abstract: Negative Poisson’s ratio, while thermodynamically permitted is only observed in some natural crystals in certain directions. All other known cases of negative Poisson’s ratio are the man-made structures exhibiting this property macroscopically. One class of these structures includes re-entrant foams and specially engineered structures with springs and hinges. Another class is constituted by materials consisting of a matrix with positive Poisson’s ratio with embedded microstructural elements producing macroscopic negative Poisson’s ratio. In the first class, we propose structures made of balls bonded by links with high shear to normal stiffness ratio. In the second class, we introduce materials filled with cracks with suppressed relative shearing of the faces. In the case of multiscale crack distributions the minimum value of Poisson’s ratio achievable is −1/3. Materials with positive and negative Poisson’s ratio can be combined into hybrid materials. We show that multiscale distribution of spherical inclusions whose Poisson’s ratio has different sign to that of the elastic isotropic matrix can considerably (up to two orders of magnitude) increase the effective Young’s modulus even when the Young’s moduli of the matrix and inclusions are the same. Generic mechanical properties of auxetics such as fracture toughness, wave propagation and damping are discussed. [Copyright &y& Elsevier]