We present some recent developments of infinite-dimensional Geometry, and show some applications in Shape Analysis. The first chapter is an introduction to the universal Teichmüller space and to its connexion to infinite-dimensional coadjoint orbits on one hand, and Shape Analysis via the fingerprint map on the other hand. Chapter II presents applications of infinite-dimensional Riemannian Geoemtry for Form Recognition and Shape Analysis of curves and surfaces. Chapter III is devoted to infinite-dimensional Poisson Geometry and its pathologies, notably we set the foundations of the theory of Banach Poisson--Lie groups.