This course introduces the mathematical foundations and modeling principles behind fluid and diffusion limits, with an emphasis on their use as approximation tools in operations. Fluid limits often capture first-order, law-of-large-numbers behavior; diffusion limits capture second-order stochastic fluctuations. These approximations are not merely technical devices: they explain congestion, delay, variability, resource pooling, heavy traffic behavior, and the qualitative structure of good controls. The course is intended for PhD students who want to develop a rigorous and reusable toolkit for stochastic modeling. While many examples will come from queues and stochastic processing systems, the methods are broadly relevant across operations, applied probability, revenue management, marketplace design, service operations, healthcare operations, and data-driven decision making. Students who take the course should leave with both a conceptual understanding of why fluid and diffusion approximations work and a technical foundation for using these methods in their own research.