The objective of this research is to model the geometric variability of the glenoid (the "socket" component of the "ball and socket" connection of the shoulder joint) of the scapula. The model must capture the observed variability with sufficient resolution such that it informs operative and design decisions. This required the quantification of variability in landmark locations and relevant bone geometry. Landmarks were placed on the existing glenoid meshes, such that they provided enough information to represent the geometry, while being consistent across each glenoid. Additionally, the surface geometry of the glenoid vault was modeled. This required the application of existing mathematical and statistical modeling approaches, including geometric fitting, radial basis functions, and principal component analysis. The landmark identification process represented the glenoid in new manner. The work was validated against existing approaches and CT scans from 42 patients. A range of information on shoulder geometries can assist with preoperative planning, as well as implant design, for Total Shoulder Arthroplasty (TSA). Principal component analysis (PCA) was used to quantify the variability of shape across the glenoid landmarks, and synthesize new glenoid models. The process of creation of these shoulder geometries may possibly be useful for the study of other joints. The models created will help surgeons and engineers to understand the effects of osteoarthritis on bone geometry, as well as the range of variability present in healthy shoulders.