Large language models (LLMs) are reshaping automated program repair. We present a unified taxonomy that groups 62 recent LLM-based repair systems into four paradigms defined by parameter adaptation and control authority over the repair loop, and overlays two cross-cutting layers for retrieval and analysis augmentation. Prior surveys have either focused on classical software repair techniques, on LLMs in software engineering more broadly, or on subsets of LLM-based software repair, such as fine-tuning strategies or vulnerability repair. We complement these works by treating fine-tuning, prompting, procedural pipelines, and agentic frameworks as first-class paradigms and systematically mapping representative systems to each of these paradigms. We also consolidate evaluation practice on common benchmarks by recording benchmark scope, pass@k, and fault-localization assumptions to support a more meaningful comparison of reported success rates. We clarify trade-offs among paradigms in task alignment, deployment cost, controllability, and ability to repair multi-hunk or cross-file bugs. We discuss challenges in current LLM-based software repair and outline research directions. Our artifacts, including the representation papers and scripted survey pipeline, are publicly available at this https URL.