As Large Language Models (LLMs) continue to grow in both capability and cost, transferring frontier capabilities into smaller, deployable students has become a central engineering problem, and knowledge distillation remains the dominant technique for this transfer. The prevailing recipe in industrial pipelines, static imitation of teacher-generated text, carries a structural weakness that grows more severe as tasks become longer and more reasoning-intensive. Because the student is trained on flawless teacher prefixes but must generate its own at inference, small errors tend to accumulate into trajectories it has rarely been trained to recover from, and the resulting exposure bias has been shown to scale roughly with the square of sequence length. On-Policy Distillation (OPD) reorganizes the training loop around this observation by having the teacher provide feedback on what the student actually produces, with the goal of reducing the compounding term toward linear and reframing distillation as an iterative correction process rather than single-pass imitation. The resulting literature has expanded along divergence design, reward-guided optimization, and self-play, yet contributions remain scattered across the knowledge distillation, RLHF, and imitation learning communities without a unified treatment. This survey provides such a treatment. We formalize OPD as $f$-divergence minimization over student-sampled trajectories, organize the field along three design axes (what to optimize, where the signal comes from, and how to stabilize training in practice), and consolidate success conditions, recurring failure modes, and the connection between OPD and KL-constrained RL. We close with open problems that emerge from this synthesis, including distillation scaling laws, uncertainty-aware feedback, agentic distillation, and the growing overlap between knowledge distillation and RL.