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The question of how soft polymers slide against hard surfaces is of significant scientific interest, given its practical implications. Specifically, such systems commonly show interesting stick–slip dynamics, wherein the interface moves intermittently despite uniform remote loading. The year 2021 marked the 50th anniversary of the publication of a seminal paper by Adolf Schallamach ( Wear , 1971), which first revealed an intimate link between stick–slip and moving detachment waves, now called Schallamach waves. We place Schallamach’s results in a broader context and review subsequent investigations of stick–slip, before discussing recent observations of solitary Schallamach waves. This variant is not observable in standard contacts so that a special cylindrical contact must be used to quantify its properties. The latter configuration also reveals the occurrence of a dual wave—the so-called separation pulse—that propagates in a direction opposite to Schallamach waves. We show how the dual wave and other, more general, Schallamach-type waves can be described using continuum theory and provide pointers for future research. In the process, fundamental analogues of Schallamach-type waves emerge in nanoscale mechanics and interface fracture. The result is an ongoing application of lessons learnt from Schallamach-type waves to better understand these latter phenomena. This article is part of the theme issue ‘Nanocracks in nature and industry’.