This paper introduces text-driven talking avatar generation, a new task that uses text to instruct both the generation and animation of an avatar. One significant obstacle in this task is the absence of paired text and talking avatar data for model training, limiting data-driven methodologies. To this end, we present a zero-shot approach that adapts an existing 3D-aware image generation model, trained on a large-scale image dataset for high-quality avatar creation, to align with textual instructions and be animated to produce talking avatars, eliminating the need for paired text and talking avatar data. Our approach's core lies in the seamless integration of a 3D-aware image generation model (i.e., EG3D), the explicit 3DMM model, and a newly developed self-supervised inpainting technique, to create and animate the avatar and generate a temporal consistent talking video. Thorough evaluations demonstrate the effectiveness of our proposed approach in generating realistic avatars based on textual descriptions and empowering avatars to express user-specified text. Notably, our approach is highly controllable and can generate rich expressions and head poses.