Fluorescent proteins (FPs) have become ubiquitous tools for biological research and concomitantly they are intriguing molecules that are amenable to study with a wide range of experimental and theoretical tools. This perspective explores the connection between the engineering of improved FPs and basic ideas from physical chemistry that explain their properties and drive the molecular design of brighter and more photostable variants. We highlight some of the progress and the many knowledge gaps in understanding the relationship between FP brightness and photostability. We also explore some of the pertinent remaining questions and suggest ways in which physical chemists might further examine the physical basis of brightness and photostability in these systems.