In the last chapter, we reviewed the classification of compact complex surfaces, especially Kodaira’s work about elliptic surfaces and the classification of singular fibers. In this chapter, we focus on the counterpart of elliptic surfaces in differential topology, that is, genus-one Lefschetz fibrations. It played an important role in the work of Kas and Moishezon which determined the diffeomorphism types of elliptic surfaces over \(\mathbb {C} P^1\) , and was completed by the theorem of Matsumoto.