Heat conduction by phonons is a ubiquitous process that incorporates a wide range of physics and plays an essential role in applications ranging from space power generation to LED lighting. Heat conduction has been studied for over two hundred years, yet many of the microscopic details have remained unknown in most crystalline solids, including which phonon-phonon interactions are primarily responsible for thermal resistance and how heat is distributed among the broad thermal spectrum. This lack of knowledge was the result of limitations on the available tools to study heat conduction. However, recent advances in both computation and experiment are enabling an unprecedented microscopic view of thermal transport by phonons in both bulk and nanostructured crystals, from the level of atomic bonding to mesoscopic transport in complex devices. In this topical review, we examine these techniques and the microscopic insights gained into the science and engineering of heat conduction.