UniMate introduces a unified foundation model designed to synthesize articulated motion for arbitrary 3D skeletons using only a rigged asset and a text prompt, eliminating the need for per-skeleton fine-tuning or test-time optimization. To overcome the topology constraints of previous animators, the researchers developed a topology-aware diffusion transformer that incorporates skeletal structure through three distinct mechanisms: a graph-aware attention bias based on pairwise joint relations and geodesic distances, a spectral rotary position embedding that generalizes RoPE to kinematic trees via the graph Laplacian, and a global topological conditioner derived from attention-pooling the rest-pose skeleton. The model was trained on UniML3D, a newly curated dataset containing 13,006 motion sequences that encompass diverse morphologies, including bipedal, quadrupedal, avian, marine, insectoid, serpentine, and articulated rigid objects. By utilizing unified canonicalization and text pairing, UniMate demonstrates superior performance in motion quality, generalization, and efficiency compared to existing state-of-the-art baselines. The framework supports various advanced animation tasks in a zero-shot manner, such as cross-topology transfer, in-betweening, motion expansion, and text-guided editing, providing a scalable solution for animating diverse 3D assets without category-specific templates.