A bounded convex subset K of a Banach space B has normal structure if for each convex subset H of K which contains more than one point there is a point x in H which is not a diametral point of H. The concept of normal structure (introduced by Brodskii and Milman) and a strengthening of this concept called complete normal structure have been of fundamental importance in some recent investigations concerned with determining conditions on weakly compact K under which the members of any commutative family ^~ of nonexpansive mappings of K into itself have a common fixedpoint. A more thorough study of these concepts is initiated in the present paper. The theorems obtained primarily concern product spaces composed of spaces which possess normal structure.