We present AnyTalk, a novel method for generating 3D speech animations for arbitrary characters without requiring any animation data. While existing audio-driven 3D speech animation methods rely on character-specific training data or laborious rigging/re-meshing, AnyTalk circumvents these limitations by leveraging recent video diffusion models trained on extensive video datasets. We first adapt a pre-trained video diffusion model to a target character through our Character-specific Fine-tuning (CsF) technique. By fine-tuning on rendered images of the 3D character paired with zeroed-out audio embeddings (representing "no motion"), we eliminate the need for animation data while preserving the motion prior of large-scale video diffusion model. We then uplift the resulting talking-head video into a 3D speech animation by estimating blendshape parameters through a proposed optimization process. AnyTalk enables lip-synced animations across diverse face meshes and blendshape configurations, significantly reducing manual effort and data requirements. We further enhance usability by distilling AnyTalk into a streamlined network, AnyTalk_{RT}, thereby enabling real-time performance. By leveraging talking-head video generation, our method broadens access to audio-driven speech animation technology for arbitrary characters. The code is publicly available at https://serin-yoon.github.io/projects/anytalk/.