This is an overview article. In his Habilitationsvortrag, Riemann described infinite dimensional manifolds parameterizing functions and shapes of solids. This is taken as an excuse to describe convenient calculus in infinite dimensions which allows for short and transparent proofs of the main facts of the theory of manifolds of smooth mappings. Smooth manifolds of immersions, diffeomorphisms, and shapes, and weak Riemannian metrics on them are treated, culminating in the surprising fact, that geodesic distance can vanish completely for them.
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