Group schemes out of birational group laws, N\'eron models

In this note, we present the theorem of extension of birational group laws in both settings of classical varieties (Weil) and schemes (Artin). We improve slightly the original proof with a more direct construction of the group extension and the systematic use of algebraic spaces, and we discuss the separation properties of the group extension. We also explain the important application to the construction of N'eron models of abelian varieties. This note grew out of lectures given by Ariane M'ezard and the second author at the Summer School “Sch'emas en groupes” held in the CIRM (Luminy) from 29 August to 9 September, 2011.