Applications for energy absorption materials range from athletic equipment, to vehicle crumple zones, to blast protection for military vehicles and personnel. Many energy absorption structures employ stochastic foams because of their plateau-like stress-strain response that allows for the absorption of large amounts of energy at relatively low stresses over large compressive strains. Periodic lattice structures, when properly designed, provide the same capabilities as stochastic systems, but with a more tailorable response that provides potential for improved specific strength and energy absorption. The present dissertation provides an in-depth study of the pyramidal lattice: one particular periodic structure that strikes a good compromise between performance and manufacturability. Through finite element and analytical modeling, this study identifies key parameters of the geometry, boundary conditions, and parent material properties that determine the compressive stress-strain response of the structure. In conjunction with experimental investigations, these models are used to understand and determine the potential for improving the response of the as-manufactured polymeric pyramidal lattice structures through additional heat treatment and filling the lattice void-space with stochastic foam. Finally, additional models are developed to understand and predict the structural rate effects that arise from inertial stabilization of strut buckling during dynamic loading. Particular emphasis is given to the effects of yield strain and density of the parent material on failure modes and dynamic response. In addition to providing a strong basis for the design of pyramidal lattice materials, this work provides useful insight into the design of energy absorption materials in general.