Thermal transport is a ubiquitous process that plays an essential role in nearly every technological application, ranging from space power generation to consumer electronics. In many of these applications, heat is carried by phonons, or quanta of lattice vibrations. Compared to other energy carriers such as electrons or photons, the microscopic properties of thermal phonons remain remarkably poorly understood, with much of our understanding still based on semi-empirical studies from over fifty years ago. In this talk, I will describe our efforts to uncover the microscopic processes that govern thermal transport by phonons using both experiment and computation. In particular, I will describe a new experimental technique that has enabled the first direct measurements of phonon mean free paths in a wide range of crystalline solids. I will demonstrate how these insights are advancing applications ranging from thermoelectric waste heat recovery to radio astronomy.