  This survey paper begins with the description of the duality between arc
systems and ribbon graphs embedded in a punctured surface. Then we explain how
to cellularize the moduli space of curves in two different ways: using
Jenkins-Strebel differentials and using hyperbolic geometry. We also briefly
discuss how these two methods are related. Next, we recall the definition of
Witten cycles and we illustrate their connection with tautological classes and
Weil-Petersson geometry. Finally, we exhibit a simple direct argument to prove
that Witten classes are stable.
