Protein-specific large language models (ProteinLLMs) are revolutionizing protein science by enabling more efficient protein structure prediction, function annotation, and design. While existing surveys focus on specific aspects or applications, this work provides the first comprehensive overview of ProteinLLMs, covering their architectures, training datasets, evaluation metrics, and diverse applications. Through a systematic analysis of over 100 articles, we propose a structured taxonomy of state-of-the-art ProteinLLMs, analyze how they leverage large-scale protein sequence data for improved accuracy, and explore their potential in advancing protein engineering and biomedical research. Additionally, we discuss key challenges and future directions, positioning ProteinLLMs as essential tools for scientific discovery in protein science. Resources are maintained at https://github.com/Yijia-Xiao/Protein-LLM-Survey.