Recent advances in LLMs have greatly improved general-domain NLP tasks. Yet, their adoption in critical domains, such as clinical trial recruitment, remains limited. As trials are designed in natural language and patient data is represented as both structured and unstructured text, the task of matching trials and patients benefits from knowledge aggregation and reasoning abilities of LLMs. Classical approaches are trial-specific and LLMs with their ability to consolidate distributed knowledge hold the potential to build a more general solution. Yet recent applications of LLM-assisted methods rely on proprietary models and weak evaluation benchmarks. In this survey, we are the first to analyze the task of trial-patient matching and contextualize emerging LLM-based approaches in clinical trial recruitment. We critically examine existing benchmarks, approaches and evaluation frameworks, the challenges to adopting LLM technologies in clinical research and exciting future directions.