this book contains a comprehensive treatment of the kepler problem, i.e., the two body problem. it is divided into four parts. in the first part the arguments are exposed elementarily, and the properties of the problem are recovered in a purely computational way. this part is written at an undergraduate student level. in the second part a unifying point of view, originally due to the author, is presented which centers the exposition on the intrinsic group-geometrical aspects. this part requires more mathematical background which is presented in the fourth part, in particular, the basic tools of differential geometry and analytical mechanics used in the book. the third part exploits some results of the second part to give a geometrical description of the perturbation theory of the kepler problem. each of the four parts, which are to some extent independent, could form the basis for a one-semester course.