Large language models have enabled agents that reason, plan, and interact with tools and environments to accomplish complex tasks. As these agents operate over extended interaction horizons, their effectiveness increasingly depends on adapting behavior to individual users and maintaining continuity across time, giving rise to personalized LLM-powered agents. In such long-term, user-dependent settings, personalization permeates the entire decision pipeline rather than remaining confined to surface-level generation. This survey provides a capability-oriented review of personalized LLM-powered agents. We organize the literature around four interdependent components: profile modeling, memory, planning, and action execution. Using this taxonomy, we synthesize representative methods and analyze how user signals are represented, propagated, and utilized, highlighting cross-component interactions and recurring design trade-offs. We further examine evaluation metrics and benchmarks tailored to personalized agents, summarize application scenarios spanning general assistance to specialized domains, and outline future directions for research and deployment. By offering a structured framework for understanding and designing personalized LLM-powered agents, this survey charts a roadmap toward more user-aligned, adaptive, robust, and deployable agentic systems, accelerating progress from prototype personalization to scalable real-world assistants.