Materials with switchable mechanical properties are widespread in living organisms and endow many species with traits that are essential for their survival. Many of the mechanically morphing materials systems found in nature are based on hierarchical structures, which are the basis for mechanical robustness and often also the key to responsive behavior. Many "operating principles" involve cascades of events that translate cues from the environment into changes of the overall structure and/or the connectivity of the constituting building blocks at various levels. These concepts permit dramatic property variations without significant compositional changes. Inspired by the function and the growing understanding of the operating principles at play in biological materials with the capability to change their mechanical properties, significant efforts have been made toward mimicking such architectures and functions in artificial materials. Research in this domain has rapidly grown in the last two decades and afforded many examples of bioinspired materials that are able to reversibly alter their stiffness, shape, porosity, density, or hardness upon remote stimulation. This review summarizes the state of research in this field.