Given a point not on a given straight line, there is one and only one line through the point and parallel to that line; this is the form by which the Fifth Postulate is usually known. This statement has always been considered to be true since the time of Euclid, not least because it states a fact about our own space. The authors start by showing the position that the postulate occupied in Euclid’s work and ultimately its place in Western Geometry. Then they examine attempts that have been made to remove the problem, either simply by attempted proofs, or more modestly by replacing the postulate with a simpler axiom, or again by modifying the definition of parallel lines. One sees how every attempt at chasing away the problem results in it returning again at full gallop. And the problem became resolved, neither by changing the definition of the Fifth Postulate, nor by proving it, but in a third way. The non-existence of a proof was established and also, paradoxically, the impossibility of not having the Fifth Postulate. This article is aimed at teachers of mathematics for use as a means of introducing a historical perspective into the teaching of mathematics. It also contains exercises to be solved according to ancient and modern methods. The chapter ends with a bibliography which contains, in addition to the historical sources that have been used, a certain number of books recommended for further study of the subject. (ZDM/Mathdi)