Matching patients to clinical trial options is critical for identifying novel treatments, especially in oncology. However, manual matching is labor-intensive and error-prone, leading to recruitment delays. Pipelines incorporating large language models (LLMs) offer a promising solution. We conducted a systematic review of studies published between 2020 and 2025 from three academic databases and one preprint server, identifying LLM-based approaches to clinical trial matching. Of 126 unique articles, 31 met inclusion criteria. Reviewed studies focused on matching patient-to-criterion only (n=4), patient-to-trial only (n=10), trial-to-patient only (n=2), binary eligibility classification only (n=1) or combined tasks (n=14). Sixteen used synthetic data; fourteen used real patient data; one used both. Variability in datasets and evaluation metrics limited cross-study comparability. In studies with direct comparisons, the GPT-4 model consistently outperformed other models, even finely-tuned ones, in matching and eligibility extraction, albeit at higher cost. Promising strategies included zero-shot prompting with proprietary LLMs like the GPT-4o model, advanced retrieval methods, and fine-tuning smaller, open-source models for data privacy when incorporation of large models into hospital infrastructure is infeasible. Key challenges include accessing sufficiently large real-world data sets, and deployment-associated challenges such as reducing cost, mitigating risk of hallucinations, data leakage, and bias. This review synthesizes progress in applying LLMs to clinical trial matching, highlighting promising directions and key limitations. Standardized metrics, more realistic test sets, and attention to cost-efficiency and fairness will be critical for broader deployment.