Summary: We show that certain right-invariant metrics endow the infinite-dimensional Lie group of all smooth orientation-preserving diffeomorphisms of the circle with a Riemannian structure. The study of the Riemannian exponential map allows us to prove infinite-dimensional counterparts of results from classical Riemannian geometry: The Riemannian exponential map is a smooth local diffeomorphism and the length-minimizing property of the geodesics holds.