Staircase profiles of temperature and composition are ubiquitously observed on Earth, including the ocean, volcanic lakes, and sedimentary flows for instance. This observation begs the question of what causes layer formation, and what role the layers play in heat and compositional transport in these natural systems. Coming from an astrophysical background, I have also been interested in the possibility that such layering might be present in stars and planets, even though they cannot be directly observed. In this review talk, I will therefore review and discuss current knowledge about layering instabilities in both geophysical and astrophysical fluid dynamics.