Time series analysis underpins critical applications in finance, healthcare, and industrial monitoring, yet traditional methods depend heavily on expert knowledge and manual intervention, limiting scalability and adaptability. With the rapid evolution of Large Language Models (LLMs), the field is shifting from static model-centric approaches toward autonomous, agent-driven systems capable of orchestrating full analytical workflows. This survey closes a crucial gap by systematically mapping this transformation through a review of over 150 studies, establishing a comprehensive taxonomy of LLM-based agent architectures grounded in five key components of the time series pipeline: perception, planning, tool use, memory, and reflection. We also identify persistent challenges such as data heterogeneity, temporal uncertainty, and latency, and outline a forward-looking research agenda emphasizing neuro-symbolic reasoning, multimodal integration, and human-agent collaboration. This work provides (1) a unified taxonomy for temporal agents, (2) comprehensive benchmark synthesis, (3) critical evaluation of existing approaches, and (4) a roadmap for future development. The continuously updated repository is available at \url{https://github.com/CoderPowerBeyond/Agent-Prompt-TS-Survey}.