Clinical trials are designed in natural language and the task of matching them to patients, represented via both structured and unstructured textual data, benefits from knowledge aggregation and reasoning abilities of LLMs. LLMs with their ability to consolidate distributed knowledge hold the potential to build a more general solution than classical approaches that employ trial-specific heuristics. Yet, adoption of LLMs in critical domains, such as clinical research, comes with many challenges, such as, the availability of public benchmarks, the dimensions of evaluation and data sensitivity. In this survey, we contextualize emerging LLM-based approaches in clinical trial recruitment. We examine the main components of the clinical trial recruitment process, discuss existing challenges in adopting LLM technologies in clinical research and exciting future directions.