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Topology optimization is increasingly being used to design the architecture of porous cellular materials with extreme elastic properties. Herein, we look to extend the design problem to the nonlinear regime and aim to maximize the energy absorption capacity until failure of the base solid occurs locally. This results in a problem formulation where the nonlinear properties are estimated using a finitely periodic structure. An interesting base material choice for energy absorption are bulk metallic glasses for which we optimize the designs and fabricate them through a thermoplastic processing method. Testing to full densification reveals that the governing mechanisms for these topologically-optimized structures are combinations of buckling and yielding at the strut-level. As a consequence, they offer superior total energy absorption over the traditional honeycomb topologies. Investigations of the same topologies made of polyether ether ketone suggest future directions on how to improve the post-peak response of topology-optimized cellular materials. [Display omitted] [ABSTRACT FROM AUTHOR]