  In this paper we review the recent advances which have taken place in the
understanding and applications of acoustic/elastic metamaterials. Metamaterials
are artificially created composite materials which exhibit unusual properties
which are not found in nature. We begin with presenting arguments from discrete
systems which support the case for the existence of unusual material properties
such as tensorial and/or negative density. The arguments are then extended to
elastic continuums through coherent averaging principles. The resulting coupled
and nonlocal homogenized relations, called the Willis relations, are presented
as the natural description of inhomogeneous elastodynamics. They are
specialized to Bloch waves propagating in periodic composites and we show that
the Willis properties display the unusual behavior which is often required in
metamaterial applications such as the Veselago lens. We finally present the
recent advances in the area of transformation elastodynamics, charting its
inspirations from transformation optics, clarifying its particular challenges,
and identifying its connection with the constitutive relations of the Willis
and the Cosserat types.
