Mechanical metamaterials are a class of engineered material designed to possess specific mechanical properties not typically found in nature. Many innovative approaches are used to obtain the desired mechanical response in these materials. For example, coefficients of thermal expansion is engineered as a means to generate specific mechanical response, such as buckling. The mechanical and thermal responses can be coupled to design novel responses such as folding and unfolding of a structure, origami inspired shapes, or tunable thermal response. In this work, various categories of metamaterials are explored such as auxetic materials with negative Poisson’s ratio, origami-inspired materials with tunable stiffness, and bandgap materials with tunable dynamic response for vibratory deformation. In addition, material with heat flux management and the use of advanced designs to control thermal transfer efficiency are discussed. Lastly, manipulation of vibratory bandgaps through structural design, blending mechanical and acoustic metamaterial design is described. Such metamaterials are extremely useful in aerospace, transportation, and robotics applications due to their tunable mechanical response and coupling of mechanical-thermal response that is capable of providing novel properties.