A classification of possible candidates for the holonomy groups of manifolds having affine connections with zero torsion discloses only groups transitive on the unit sphere in the tangent space of the manifold, except in the case where the manifold is a symm ric space of rank greater than or equal to 2. An intrinsic proof of this rather startling fact, and an algebraic generalization of the notion of a holonomy group are given with a short, intrinsic proof of the result on transitivity. Al hough only that portion of the problem which has to do wit R IEMANNIAN MANIFOLDS IS TREATED, IT IS POSSIBLE H T THE DEVICES EMPLOYED COULD BE ALTERED TO PERTAIN TO OTHER SITUATIONS. (Author).