Generative AI is reshaping art, gaming, and most notably animation. Recent breakthroughs in foundation and diffusion models have reduced the time and cost of producing animated content. Characters are central animation components, involving motion, emotions, gestures, and facial expressions. The pace and breadth of advances in recent months make it difficult to maintain a coherent view of the field, motivating the need for an integrative review. Unlike earlier overviews that treat avatars, gestures, or facial animation in isolation, this survey offers a single, comprehensive perspective on all the main generative AI applications for character animation. We begin by examining the state-of-the-art in facial animation, expression rendering, image synthesis, avatar creation, gesture modeling, motion synthesis, object generation, and texture synthesis. We highlight leading research, practical deployments, commonly used datasets, and emerging trends for each area. To support newcomers, we also provide a comprehensive background section that introduces foundational models and evaluation metrics, equipping readers with the knowledge needed to enter the field. We discuss open challenges and map future research directions, providing a roadmap to advance AI-driven character-animation technologies. This survey is intended as a resource for researchers and developers entering the field of generative AI animation or adjacent fields. Resources are available at: https://github.com/llm-lab-org/Generative-AI-for-Character-Animation-Survey.