This thesis presents the investigation of lattice structures for the purpose of providing novel pathways for wave and mechanical property control. Structures of the type studied are often called metamaterials or metastructures as they behave in a way that is beyond their constituents. Magneto-elastic lattices are the primary focus, since they are generally multistable, allowing one structure to assume various geometric configurations, which can correspond to various functional modes, providing the possibility for adaptive structures. Periodic structures are considered for their unique wave properties, as well as for ease of design and application. First, possible configurations are identified. Then, dynamic reconfiguration is investigated as a fast and versatile method for switching configurations. Bloch wave analysis is applied to magneto-elastic lattices to investigate the introduction of anisotropic wave propagation, opening of bandgaps, and changes in wave speeds due to geometric lattice reconfiguration. The equivalent continuum properties of the lattice structures are also calculated, using a homogenization methodology. Reconfiguration from hexagonal to re-entrant lattices produces over an order of magnitude change in stiffness while converting the lattice from having isotropic to orthotropic properties. This thesis also addresses two other topics relevant to the development of periodic structures that seek to control wave energy. The first, pursuing novel wave control, is topologically protected edge mode. Building by analogy from quantum mechanics, elastic mechanical structures are designed that carry special waves modes in only one direction along a lattice boundary. Another topic is the development of a methodology for measuring in-plane waves in lattice structures. As a new methodology, digital image correlation is combined with high-speed photography.