This paper provides the first and most comprehensive analysis of the convergence of Web3 technologies and AI agents. We examine the intersection of Web3 and AI agents across five key dimensions: landscape, economy, governance, security, and trust mechanisms. Through the analysis of 133 existing projects, we develop a taxonomy that identifies distinct patterns in project distribution and capitalization and systematically map the current market landscape (RQ1). Building on these findings, we further investigate four key integrations: (1) the role of AI agents in participating in and optimizing decentralized finance (DeFi) (RQ2), (2) their contribution to enhancing Web3 governance mechanisms (RQ3), (3) their ability to enhance Web3 security through intelligent vulnerability detection and automated smart contract auditing (RQ4), and (4) building a robust trust framework for AI agent operation leveraging Web3's unique trust infrastructure (RQ5). By synthesizing these dimensions, we identify key integration patterns, highlight fundamental challenges related to scalability, security, and ethics, and outline important considerations for future research aimed at building robust, intelligent, and trustworthy decentralized systems through effective AI agent interaction.