We develop a general framework for the study of a class of randomized load balancing models in networks with many servers and in the presence of general service distributions. A key ingredient is a convenient Markovian representation of the state in terms of interacting measure-valued processes. We apply the framework to establish mean-field limits for randomized load balancing models using the so-called "power of two choices" scheduling policy. We also identify a system of integro-differential equations that provide a dynamic description of key performance measures in the system and describe some surprising engineering insights that can be gleaned from this analysis. This is based on joint works with Mohammadreza Aghajani and Xingjie (Helen) Li.