Mumford and Thurston observed that the theory of Jenkins-Strebel differentials can be used to triangulate a compactification of the moduli space of n-pointed Riemann surfaces of given genus. This was later exploited by Kontsevich to prove the Witten conjecture. We review this construction, make a comparison with the Deligne-Mumford compacfication and discuss some of the conjectures in algebraic geometry it led to.