After extensive practice with motor tasks sharing structural similarities (e.g., different dancing movements, or different sword techniques), new tasks of the same type can be learned faster. According to the recent "structure learning" hypothesis (Braun et al., 2009a), such rapid generalization of related motor skills relies on learning the dynamic and kinematic relationships shared by this set of skills. As a consequence, motor adaptation becomes constrained, effectively leading to a dimensionality reduction of the learning problem; at the same time, adaptation to tasks lying outside the structure becomes biased toward the structure. We tested these predictions by investigating how previously learned structures influence subsequent motor adaptation. Human subjects were making reaching movements in 3D virtual reality, experiencing perturbations either in the vertical or in the horizontal plane. Perturbations were either visuomotor rotations of varying angle or velocity-dependent forces of varying strength. We found that, after extensive training with both kinematic or dynamic perturbations, adaptation to unpracticed, diagonal, perturbations happened along the previously learned structure (vertical or horizontal), and resulting adaptation trajectories were curved. This effect is robust, can be observed on the single-subject level, and occurs during adaptation both within and across trials. Additionally, we demonstrate that structure learning changes involuntary visuomotor reflexes and therefore is not exclusively a high-level cognitive phenomenon.