A golf club, particularly a “putter” or “chipper”, is provided with a concave ball striking face that establishes line contact with a golf ball rather than the usual point contact and enhances the capability of a user to control the direction of ball movement resulting from a golf stroke. The concave ball striking surface may be curved and may have the same or greater curvature than the curvature of a golf ball and may have smoothly and gently merging surface regions of compound curvature. The concave ball striking face surface may have compound regions including a curved region and a planar region being smoothly and gently merged. The concave ball striking surface may be defined on an integral metal club head structure or may be defined by a polymer insert that is recessed into an insert receiving receptacle of the club head. Additionally, the face portion of a golf club head may have a recessed face to further aid in training and to enhance the efficiency and accuracy of ball movement resulting from a golf stroke.