Time series analysis is pivotal in domains like financial forecasting and biomedical monitoring, yet traditional methods are constrained by limited nonlinear feature representation and long-term dependency capture. The emergence of Large Language Models (LLMs) offers transformative potential by leveraging their cross-modal knowledge integration and inherent attention mechanisms for time series analysis. However, the development of general-purpose LLMs for time series from scratch is still hindered by data diversity, annotation scarcity, and computational requirements. This paper presents a systematic review of pre-trained LLM-driven time series analysis, focusing on enabling techniques, potential applications, and open challenges. First, it establishes an evolutionary roadmap of AI-driven time series analysis, from the early machine learning era, through the emerging LLM-driven paradigm, to the development of native temporal foundation models. Second, it organizes and systematizes the technical landscape of LLM-driven time series analysis from a workflow perspective, covering LLMs' input, optimization, and lightweight stages. Finally, it critically examines novel real-world applications and highlights key open challenges that can guide future research and innovation. The work not only provides valuable insights into current advances but also outlines promising directions for future development. It serves as a foundational reference for both academic and industrial researchers, paving the way for the development of more efficient, generalizable, and interpretable systems of LLM-driven time series analysis.