Recent advancements in instructing Large Language Models (LLMs) to utilize external tools and execute multi-step plans have significantly enhanced their ability to solve intricate tasks, ranging from mathematical problems to creative writing. Yet, there remains a notable gap in studying the capacity of LLMs in responding to personalized queries such as a recommendation request. To bridge this gap, we have designed an LLM-powered autonomous recommender agent, RecMind, which is capable of providing precise personalized recommendations through careful planning, utilizing tools for obtaining external knowledge, and leveraging individual data. We propose a novel algorithm, Self-Inspiring, to improve the planning ability of the LLM agent. At each intermediate planning step, the LLM 'self-inspires' to consider all previously explored states to plan for next step. This mechanism greatly improves the model's ability to comprehend and utilize historical planning information for recommendation. We evaluate RecMind's performance in various recommendation scenarios, including rating prediction, sequential recommendation, direct recommendation, explanation generation, and review summarization. Our experiment shows that RecMind outperforms existing zero/few-shot LLM-based recommendation methods in different recommendation tasks and achieves competitive performance to a recent model P5, which requires fully pre-train for the recommendation tasks.