Display omitted Mechanical metamaterials are man-made structures with counterintuitive mechanical properties that originate in the geometry of their unit cell instead of the properties of each component. The typical mechanical metamaterials are generally associated with the four elastic constants, the Young's modulus E, shear modulus G, bulk modulus K and Poisson's ratio υ, the former three of which correspond to the stiffness, rigidity, and compressibility of a material from an engineering point of view. Here we review the important advancements in structural topology optimisation of the underlying design principles, coupled with experimental fabrication, thereby to obtain various counterintuitive mechanical properties. Further, a clear classification of mechanical metamaterials have been established based on the fundamental material mechanics. Consequently, mechanical metamaterials can be divide into strong-lightweight (E/ρ), pattern transformation with tunable stiffness, negative compressibility (−4G/3 < K < 0), Pentamode metamaterials (G ≪ K) and auxetic metamaterials (G ≫ K), simultaneously using topology optimisation to share various fancy but feasible mechanical properties, ultralight, ultra-stiffness, well-controllable stiffness, vanishing shear modulus, negative compressibility and negative Poisson’s ratio. We provide here a broad overview of significant potential mechanical metamaterials together with the upcoming challenges in the intriguing and promising research field.