Motion retargeting seeks to faithfully replicate the spatio-temporal motion characteristics of a source character onto a target character with a different body shape. Apart from motion semantics preservation, ensuring geometric plausibility and maintaining temporal consistency are also crucial for effective motion retargeting. However, many existing methods prioritize either geometric plausibility or temporal consistency. Neglecting geometric plausibility results in interpenetration while neglecting temporal consistency leads to motion jitter. In this paper, we propose a novel sequence-to-sequence model for seamless Spatial-Temporal aware motion Retargeting (STaR), with penetration and consistency constraints. STaR consists of two modules: (1) a spatial module that incorporates dense shape representation and a novel limb penetration constraint to ensure geometric plausibility while preserving motion semantics, and (2) a temporal module that utilizes a temporal transformer and a novel temporal consistency constraint to predict the entire motion sequence at once while enforcing multi-level trajectory smoothness. The seamless combination of the two modules helps us achieve a good balance between the semantic, geometric, and temporal targets. Extensive experiments on the Mixamo and ScanRet datasets demonstrate that our method produces plausible and coherent motions while significantly reducing interpenetration rates compared with other approaches.