The emergence of additive manufacturing in combination with the advancement of engineering analysis tools has led to a new paradigm in the design of materials, in which the organization of matter and topology plays a central role. A new class of artificial materials has emerged that exhibit static and dynamic properties typically not encountered in natural materials and have been named as metamaterials. In the current work, we present recent advances made in the field, relating to both the static attributes and to the wave propagation characteristics of metamaterials under small and large strains. In particular, we present a class of anti-auxetic inner material architectures with positive and high Poissons ratio values which lead to metamaterials of controlled anisotropy. Thereupon, we study the influence of the degree of anisotropy on the wave propagation characteristics under small strains. Moreover, we analyze the effect of geometrical nonlinearities on the propagation of longitudinal and shear waves in two-dimensional hexagonal-shaped architectured lattices viewed as effective 1D and 2D media, while we showcase the potential of non-auxetic architectures to reach auxetic ones through a kinematic control in the large strains regime. What is more, we analyze the role of the considered nonlinearity in the nature of propagating waves, identifying strain thresholds beyond which both supersonic and subsonic modes emerge.