<p>In the digital era, stories no longer spread solely as discrete facts or isolated data points but as rich, interpretive narratives that evolve and resonate across social media platforms. This survey systematically examines the emerging field of narrative diffusion, the process by which emotionally charged, coherent stories propagate through online networks, and contrasts it with traditional models of information diffusion. We begin by reviewing foundational models, such as epidemiological and cascade-based frameworks (e.g., SI/SIR, Independent Cascade, Linear Threshold, and their extensions for temporal, topical, and neural modeling of static content). We then introduce a complementary taxonomy of narrative-specific approaches, including narrative tracking and evolution models, role-based event chains, multimodal variational methods, stance-aware epidemic adaptations, and cross-platform coordination frameworks. After reviewing both the models for information and narrative diffusion, we compare and contrast how these models can be adopted for narrative diffusion and what further can be done. By mapping these models onto the distinct structural, semantic, and emotional dimensions of narratives, we identify core challenges, such as semantic drift, multimodal integration, lack of annotated corpora, and ethical risks, and highlight gaps in evaluation practices. Finally, we outline future directions toward unified, multimodal, and cross-platform narrative diffusion. This survey aims to equip researchers with a comprehensive conceptual and methodological foundation for advancing narrative-aware analysis and intervention in online discourse.</p>