Differential Geometry is often considered as a "classical" field in Mathematics. Yet it has recently been the object of a substantial renewed interest, thanks in particular to various applications where it plays an essential role.

After a quick review of some basic notions of Differential Geometry, such as the fundamental forms of a surface or the fundamental theorem of surface theory, some applications - old and new - will be likewise briefly reviewed, such as cartography, the theory of elastic shells, or the optimization of the shape of gears.

This lecture is intended for undergraduate and graduate students. No a priori knowledge of Differential Geometry will be assumed.