Joint prosthesis with infinitely positionable head

A joint prosthesis includes a bone engaging portion, such as a stem, and articulating component, such as a humeral head, and an insert component that supports the articulating component and is removable from the stem. A mating component mates with the humeral head and is configured to engage the insert component at adjustable angles according to the anatomy of the patient's joint. In a revision surgery, the stem remains within the patient's bone while the insert component is removed with the mating component and humeral head fixed to the insert component. A new insert component may be prepared with the mating component and humeral head oriented to replicate the angular orientation of the previous prosthesis. The new insert component is then mounted within an insert cavity in the implanted stem and rigidly fastened to complete the revision.