  The ``Painlev\'e analysis'' is quite often perceived as a collection of
tricks reserved to experts. The aim of this course is to demonstrate the
contrary and to unveil the simplicity and the beauty of a subject which is in
fact the theory of the (explicit) integration of nonlinear differential
equations.
  To achieve our goal, we will not start the exposition with a more or less
precise ``Painlev\'e test''. On the contrary, we will finish with it, after a
gradual introduction to the rich world of singularities of nonlinear
differential equations, so as to remove any cooking recipe.
  The emphasis is put on embedding each method of the test into the well known
theorem of perturbations of Poincar\'e. A summary can be found at the beginning
of each chapter.
