In this expository article written for a non-specialist public, I should like to relate about present day picture of the lagrange and Hamilton geometries applied in Mechanics, Physics, etc. Of course, the subject is not a simple one and the attempt to humanize it is a difficult operation. This is the reason because I divided the matter in two parts. One is devoted to an elementary exposition of these geometrical models. The second part addresses to the graduate level. All problems discussed here can be found in the books [1], [2], [4], [5], [6].