The moduli space of curves Mg,n which parameterizes isomorphism classes of finite type Riemann surfaces is an object of study in many distinct areas of Mathematics. In this thesis we access this fundamental object through two different viewpoints: Teichmüller theory and Lefschetz fibrations. Both of these perspectives are tied to the study of maps from and to Mg,n. This thesis consists of four projects. The first two, one in collaboration with Frederik Benirschke, study the rigidity of maps to Mg,n arising from classical constructions in algebraic geometry. The last two, in collaboration with Seraphina Lee, study the flexibility of symplectic Lefschetz fibrations in contrast to their classical algebro-geometric counterparts. A key component of our work is to realize the fruitful interactions between the topological, holomorphic and symplectic structures naturally present in the objects of study.
