Yielding and gelling soft matter materials are ubiquitous throughout biological and geological systems, the most commonly encountered examples being in food and other household products. In this Highlight, we attempt to summarise the rheological properties that are characteristic of structured soft matter systems, including their universal flow behaviour and viscoelastic response as well as appropriate methods of characterisation. We also discuss how the mechanical response depends on the materials’ microstructure. Soft matter of this nature is typically in a non-equilibrium state, which means that it can be modified during processing and storage (aging) to obtain new structural states and rheologies. Several universal features have recently been observed experimentally for the linear and non-linear response of structured soft matter, and such developments are assisting in the development of suitable models to characterise their behaviour.