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This article gives an introduction to certain aspects of the asymptotic geometry of metric spaces. Thus the focus is on the large scale geometry of a space while the local structure is neglected. In particular hyperbolic spaces (in the sense of Gromov) are discussed, for which the asymptotic geometry is encoded in the boundary at infinity. In analogy with the classical hyperbolic space and its boundary, the relation between the metric geometry of a Gromov hyperbolic space and the Möbius geometry of its boundary is studied in detail.