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Résumé : This will be a description of a few highlights in the early history of non-euclidean geometry, and a few miscellaneous recent developments.An Appendix describes some explicit formulas concerning volume in hyperbolic 3-space.The mathematical literature on non-euclidean geometry begins in 1829 with publications by N. Lobachevsky in an obscure Russian journal.The infant subject grew very rapidly.Lobachevsky was a fanatically hard worker, who progressed quickly from student to professor to rector at his university of Kazan, on the Volga.Already in 1829, Lobachevsky showed that there is a natural unit of distance in non-euclidean geometry, which can be characterized as follows.In the right triangle of Figure 1 with fixed edge a, as the opposite vertex A moves infinitely far away, the angle 9 will increase to a limit 9 0 which is assumed to be strictly less than 7r/2.He showed that a = -log tan(0 o /2) if the unit of distance is suitably chosen.In particular, a « (TT/2) -0 O if a is very small.(In the interpretation introduced by Beltrami forty years later, this unit of distance is chosen so that curvature = -1.)