Cork, a natural material from renewable resources, is currently attracting increasing interest in different industrial fields because of its cellular structure and the presence of the flexible suberin as its main chemical component. In an agglomerated form, it proved to be a compelling product not only as a thermal and acoustic insulator, but also as core material in sandwich structures and as a liner or padding in energy absorbing equipment. From this perspective, the assessment of its compressive response is fundamental to ensure the right out-of-plane stiffness required to a core material and the proper crashworthiness in the safety devices. Considering the complex nature of cork and the resulting peculiar compressive response, the present review article provides an overview of this paramount property, assessing the main parameters (anisotropy, temperature, strain rate, etc.) and the peculiar features (near-zero Poisson’s ratio and unique dimensional recovery) that characterize it in its natural state. Furthermore, considering its massive exploitation in the agglomerated form, the design parameters that allow its compressive behavior to be tailored and the operating parameters that can affect its crashworthiness were assessed, reporting some potential industrial applications.