Storyboarding is widely used for creating 3D animations. Animators use the 2D sketches in storyboards as references to craft the desired 3D animations through a trial-and-error process. The traditional approach requires exceptional expertise and is both labor-intensive and time-consuming. Consequently, there is a high demand for automated methods that can directly translate 2D storyboard sketches into 3D animations. This task is under-explored to date and inspired by the significant advancements of motion diffusion models, we propose to address it from the perspective of conditional motion synthesis. We thus present Sketch2Anim, composed of two key modules for sketch constraint understanding and motion generation. Specifically, due to the large domain gap between the 2D sketch and 3D motion, instead of directly conditioning on 2D inputs, we design a 3D conditional motion generator that simultaneously leverages 3D keyposes, joint trajectories, and action words, to achieve precise and fine-grained motion control. Then, we invent a neural mapper dedicated to aligning user-provided 2D sketches with their corresponding 3D keyposes and trajectories in a shared embedding space, enabling, for the first time, direct 2D control of motion generation. Our approach successfully transfers storyboards into high-quality 3D motions and inherently supports direct 3D animation editing, thanks to the flexibility of our multi-conditional motion generator. Comprehensive experiments and evaluations, and a user perceptual study demonstrate the effectiveness of our approach.