Large Language Models (LLMs) are increasingly deployed as agentic systems that plan, memorize, and act in open-world environments. This shift brings new security problems: failures are no longer only unsafe text generation, but can become real harm through tool use, persistent memory, and interaction with untrusted web content. In this survey, we provide a transition-oriented view from Secure Agentic AI to a Secure Agentic Web. We first summarize a component-aligned threat taxonomy covering prompt abuse, environment injection, memory attacks, toolchain abuse, model tampering, and agent network attacks. We then review defense strategies, including prompt hardening, safety-aware decoding, privilege control for tools and APIs, runtime monitoring, continuous red-teaming, and protocol-level security mechanisms. We further discuss how these threats and mitigations escalate in the Agentic Web, where delegation chains, cross-domain interactions, and protocol-mediated ecosystems amplify risks via propagation and composition. Finally, we highlight open challenges for web-scale deployment, such as interoperable identity and authorization, provenance and traceability, ecosystem-level response, and scalable evaluation under adaptive adversaries. Our goal is to connect recent empirical findings with system-level requirements, and to outline practical research directions toward trustworthy agent ecosystems.

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