Starting from a short review of Riemann's sketch of a geometry on manifolds (1854ff.) this talk discusses how Hermann Weyl proposed to take up Riemann's ideas and how to modify them in the light of his views of the new developments in the foundations of analysis and in physics (the general theory of relativity, GTR). Although in his early work (1913) Weyl contributed to the later axiomatisation of the concept of manifolds, his own preferences lay clearly in a constructive approach (for a time even an intuitionistic one). Moreover, he saw a basic problem for accepting Riemannian geometry as an adequate geometric framework for a field theoretic foundation of physics. As an alternative he proposed a ``purely infinitesimal'' approach to the metric -- later called Weyl geometry. It contained the first instance of an (explicit) gauge structure which --in transformed form -- became the grandfather of the later gauge theories in physics.