The rapid expansion of sixth-generation (6G) wireless networks and the Internet of Things (IoT) has catalyzed the evolution from centralized cloud intelligence towards decentralized edge general intelligence. However, traditional edge intelligence methods, characterized by static models and limited cognitive autonomy, fail to address the dynamic, heterogeneous, and resource-constrained scenarios inherent to emerging edge networks. Agentic artificial intelligence (Agentic AI) emerges as a transformative solution, enabling edge systems to autonomously perceive multimodal environments, reason contextually, and adapt proactively through continuous perception-reasoning-action loops. In this context, the agentification of edge intelligence serves as a key paradigm shift, where distributed entities evolve into autonomous agents capable of collaboration and continual adaptation. This paper presents a comprehensive survey dedicated to Agentic AI and agentification frameworks tailored explicitly for edge general intelligence. First, we systematically introduce foundational concepts and clarify distinctions from traditional edge intelligence paradigms. Second, we analyze important enabling technologies, including compact model compression, energy-aware computing strategies, robust connectivity frameworks, and advanced knowledge representation and reasoning mechanisms. Third, we provide representative case studies demonstrating Agentic AI's capabilities in low-altitude economy networks, intent-driven networking, vehicular networks, and human-centric service provisioning, supported by numerical evaluations. Furthermore, we identify current research challenges, review emerging open-source platforms, and highlight promising future research directions to guide robust, scalable, and trustworthy Agentic AI deployments for next-generation edge environments.