Frictionally yielding media are a particular type of non-Newtonian fluids that significantly deform under stress and do not recover their original shape. For example, mud, snow, soil, leaf litters, or sand are such substrates because they flow when stress is applied but do not bounce back when released. Some robots have been designed to move on those substrates. However, compared to moving on solid ground, significantly fewer prototypes have been developed and only a few prototypes have been demonstrated outside of the research laboratory. This paper surveys the existing biology and robotics literature to analyze principles of physics facilitating motion on yielding substrates. We categorize animal and robot locomotion based on the mechanical principles and then further on the nature of the contact: discrete contact, continuous contact above the material, or through the medium. Then, we extract different hardware solutions and motion strategies enabling different robots and animals to progress. The result reveals which design principles are more widely used and which may represent research gaps for robotics. We also discuss that higher level of abstraction helps transferring the solutions to the robotics domain also when the robot is not explicitly meant to be bio-inspired. The contribution of this paper is a review of the biology and robotics literature for identifying locomotion principles that can be applied for future robot design in yielding environments, as well as a catalog of existing solutions either in nature or man-made, to enable locomotion on yielding grounds.