Continual learning (CL) has emerged as a pivotal paradigm to enable large language models (LLMs) to dynamically adapt to evolving knowledge and sequential tasks while mitigating catastrophic forgetting-a critical limitation of the static pre-training paradigm inherent to modern LLMs. This survey presents a comprehensive overview of CL methodologies tailored for LLMs, structured around three core training stages: continual pre-training, continual fine-tuning, and continual alignment.Beyond the canonical taxonomy of rehearsal-, regularization-, and architecture-based methods, we further subdivide each category by its distinct forgetting mitigation mechanisms and conduct a rigorous comparative analysis of the adaptability and critical improvements of traditional CL methods for LLMs. In doing so, we explicitly highlight core distinctions between LLM CL and traditional machine learning, particularly with respect to scale, parameter efficiency, and emergent capabilities. Our analysis covers essential evaluation metrics, including forgetting rates and knowledge transfer efficiency, along with emerging benchmarks for assessing CL performance. This survey reveals that while current methods demonstrate promising results in specific domains, fundamental challenges persist in achieving seamless knowledge integration across diverse tasks and temporal scales. This systematic review contributes to the growing body of knowledge on LLM adaptation, providing researchers and practitioners with a structured framework for understanding current achievements and future opportunities in lifelong learning for language models.