Subcellular peroxisome numbers are balanced between formation and loss. Peroxisome degradation in mammalian cells primarily occurs through specific autophagy, involving the accumulation of receptor proteins. In turn, receptors can be recruited by surface ubiquitin. I will present quantitative models for ubiquitin and receptor cluster dynamics on peroxisome surfaces. Peroxisome protein import is guided by recycling shuttle proteins, which are labelled by ubiquitin during the import cycle. The mechanism of protein translocation across the peroxisome membrane, and the coupling of translocation to shuttle protein export, are speculative. We have developed a quantitative model for the import process using diffusion-limited rates. We find that changes to the coupling scheme cause qualitative differences in ubiquitin accumulation, and propose a natural disuse signal that could enhance autophagy. Autophagy receptor proteins have been observed in clusters, and the primary receptor protein for peroxisome autophagy, NBR1, has domains that suggest clustering behaviour. We model NBR1 dynamics on peroxisome membranes as coarsening clusters on drops. We recover standard coarsening cluster growth and scaling, but find that the scaling function varies with drop polydispersity. We also find that clusters evaporate from smaller drops and grow on larger drops, suggesting peroxisomes could be selected for autophagy based on size. Work currently underway, connecting ubiquitin accumulation and receptor clustering, will be mentioned if time allows.