Uniqueness theorems for variational problems by the method of transformation groups A classical problem in the calculus of variations is the investigation ofcritical points of functionals {\\cal L} on normed spaces V. The present work addresses the question: Under what conditions on the functional {\\cal L} and the underlying space V does {\\cal L} have at most one critical point A sufficient condition for uniqueness is given: the presence of a variational sub symmetry i.e. a one parameter group G of transformations of V which strictly reduces the values of {\\cal L}. The method of transformation groups is applied to second order elliptic boundary value problems on Riemannian manifolds. Further applications include problems of geometric analysis and elasticity. TOC:Introduction. Uniqueness of Critical Points (I). Uniqueness of Citical Pints (II). Variational Problems on Riemannian Manifolds. Scalar Problems in Euclidean Space. Vector Problems in Euclidean Space. Fréchet differentiability. Lipschitz properties of ge andomegae