**Abstract** : The mathematical problem of cartography is that of finding maps that optimize distortion, in a sense that needs to be made precise. Indeed, the word ``distortion" is attached to a variety of parameters: distances, areas and angles, and the general problem is to find mappings that are closest to mappings that make a compromise between these parameters among all mappings from a given subset of the sphere onto the Euclidean plane. Furthermore, geographers usually had additional constraints on the desired maps, such as preserving distances along a certain meridian---or along all meridians, or of sending parallels (circles centered at the poles) to parallel straight lines, or to concentric circles, or to other types of ``parallel" curves in the plane. We shall mention examples of mappings from the sphere to the Euclidean plane satisfying such requirements.
The various approaches to the question of maps with minimal distortion from (subsets of) the sphere onto the Euclidean plane acted as a motivation for Euler, Lagrange, Gauss, Chebyshev and other geometers to study general maps between differentiable surfaces. Furthermore, geography gave rise to an early version of extremal quasiconformal mappings, as a particular class of least-distortion mappings. This took place in the nineteenth century, before quasiconformal mappings were officially introduced in the late 1920s as a tool in conformal geometry and before they led to the first known metric on Teichmüller space. Thurston's theory of best Lipschitz maps, developed in his paper ``Minimal stretch maps between hyperbolic surfaces", is also based on the idea of studying maps with least distortion between surfaces. It led him to the definition of another metric on Teichmüller space, the Thurston metric, which is today an active research topic. We shall discuss all this in the present paper.
We shall also see how least distortion maps occur in art, biology and the medical sciences, and in particular in brain imaging.