This talk will be an introduction to Bridgeland stability conditions on triangulated categories, oriented towards researchers in the field of quadratic differentials. After introducing the definitions and working out some explicit examples of spaces of stability conditions, I will summarize the results of Bridgeland-Smith and Haiden-Katzarkov-Kontsevich, which are two different, but related, appearances of quadratic differentials in the study of stability conditions. Time allowing I will discuss some of my results on the subject.