Entropy mechanism emerges as a key organizing principle for understanding and improving Large Reasoning Models (LRMs). This survey examines how entropy shapes both their training and inference behavior. On the training side, we take a mathematical perspective: casting existing RL-based reasoning algorithms into a unified objective and using this formulation to derive an entropy-centric decomposition of methods, clarifying how different approaches adjust exploration–exploitation "knobs". On the inference side, based on the characteristic of LRM that trading increased inference tokens yields better performance, we summarize methods that leverage entropy to enhance inference performance or reduce uncertainty. Finally, we discuss open challenges and future directions of entropy-driven research for LRMs. Our repository is available on https://anonymous.4open.science/r/Awesome_LRM_with_Entropy-0503.