Appears in collection : Arithmetic, Geometry, Cryptography and Coding Theory / Arithmétique, géométrie, cryptographie et théorie des codes

We discuss various explicit models of genus one curves, some classical and some a little less so, with an eye towards applications in number theory and arithmetic geometry. In particular, we will talk about how understanding such models has shed light on many kinds of problems, such as computing and bounding rational (and integral) points on elliptic curves, the Hasse principle, splitting Brauer classes, and classical geometric constructions.