Here is the first part of a work that provides a full account of Jorgensen's theory of
punctured torus Kleinian groups and its generalization. It offers an elementary and
self-contained description of Jorgensen's theory with a complete proof. Through various
informative illustrations readers are naturally led to an intuitive synthetic grasp of the
theory which clarifies how a very simple fuchsian group evolves into complicated Kleinian
groups.
