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Wave absorption metamaterials have been an enduring topic over the past two decades, propelled not only by novel scientific advances, but also by their extensive application potential. In this review, we aim to provide some general insights into the absorption mechanism common to both microwave and acoustic systems. By establishing a universal model for resonance-based metamaterials, we present the theoretical conditions for broadband impedance matching and introduce the fundamental causal limit as an evaluation tool for absorption performance. Under this integrated framework, we survey the recent advances on metamaterials absorption in both microwave and acoustic systems, with the focus on those that pushed the overall performance close to the causal limit. We take note of some new, emerging metastructures that can circumvent the constraint imposed by causal limit, thereby opening a new avenue to low-frequency absorption. This review concludes by discussing the existing challenges with possible solutions and the broad horizon for future developments.