Over the last decades, Lee-Yang theory has developed into an indispensable tool to describe and understand phase transitions in equilibrium statistical physics. However, surprisingly little attention has been devoted to extending the Lee-Yang theory to phase transitions beyond classical equilibrium phase transitions. Motivated by recent experiments, the aim of this project is to develop a theory of phase transitions in interacting quantum many-body systems. We will connect this theoretical framework to other key concepts in quantum transport, quantum information, and quantum thermodynamics. With this theory at hand, we will be able to predict the large-scale phase behavior of future engineered quantum and nanoscale devices, such as quantum computers, nanomaterials, and nanomedicines.