The Prékopa-Leindler inequality (PL) and its strengthening, the Borell-Brascamp-Lieb inequality, are functional extensions of the Brunn-Minkowski inequality from convex geometry, which itself refines the classical isoperimetric inequality. These fundamental inequalities play a central role in geometry, analysis, and probability, motivating the significant attention their stability has garnered in recent years. While significant progress has been made in the geometric setting, a stability results for the Prékopa-Leindler inequality have remained elusive. In this talk, we will explore these inequalities and present recent results that resolve the long-standing question of their quantitative stability. Based on joint work with Alessio Figalli and Marius Tiba.