Available LLMs differ substantially in cost, latency, and capability. Their strengths vary across queries, so the natural question becomes: which model for which query? This is the LLM routing problem. Routing is already used in production, and the academic literature has grown across cascading, bandit, generative, and benchmark threads, each with its own vocabulary, each largely ignoring the others. We argue the field is best read along a single axis: the algorithmic operation the router performs. We organise the recurring operations under four labels. Predictive routers score candidates before their answers are generated. Adaptive routers try a model, observe the outcome, and adjust. Generative routers let an LLM itself reason about the decision. Infrastructure work studies routing rather than performs it, via benchmarks and adversarial analysis. This survey formalises the routing decision under this four-paradigm lens, traces the methodological development of the field, compares assumptions and tradeoffs across paradigms, and identifies open research directions revealed by the taxonomy. A continuously updated reading list lives at https://github.com/wuxingyu-ai/LLM-Routing.