Many existing exoskeletons have followed a similar design concept that a rigid kinematic chain is actuated to mobilize a human wearer in spite of the intended applications. For performance-augmenting applications where an exoskeleton is usually paired with a specific wearer, the human–machine kinematic compatibility might be well maintained. However, in a clinical setting for rehabilitation where one exoskeleton is often shared by a group of patients, it will be difficult for the therapists to guarantee the on-site adjustments would accurately fit the exoskeleton to each individual patient with his/her unique anatomy. This paper proposes a continuum shoulder exoskeleton design to realize anatomy adaptive assistances (AAAs) for hemiparetic patients in a purely assistive mode where patient's limb motions are passive. The shoulder exoskeleton conforms to distinct human anatomies adaptively due to its intrinsic flexibility but still manages to deliver motion assistances in a consistent way. The design concept and the system descriptions are elaborated, including kinematics, statics, system construction, actuation, experimental validation, backbone shape identification, motion compensation, manikin trials, etc. The results suggest that it is possible to design a continuum exoskeleton to assist different patients with their limb movements, while no mechanical adjustments on the exoskeleton shall be performed.