  This paper presents a survey of recent and not so recent results concerning
the study of smooth homeomorphisms of the circle with a finite number of
non-flat critical points, an important topic in the area of One-dimensional
Dynamics. We focus on the analysis of the fine geometric structure of orbits of
such dynamical systems, as well as on certain ergodic-theoretic and
complex-analytic aspects of the subject. Finally, we review some conjectures
and open questions in this field.
