Inserted: 12 may 2016
Last Updated: 12 may 2016
survey written for a summer school on metric spaces
Carnot groups are distinguished spaces that are rich of structure: they are those Lie groups equipped with a path distance that is invariant by left-translations of the group and admit automorphisms that are dilations with respect to the distance. We present the basic theory of Carnot groups together with several remarks. We consider them as special cases of graded groups and as homogeneous metric spaces. We discuss the regularity of isometries in the general case of Carnot-Carath\'eodory spaces and of nilpotent metric Lie groups.