This chapter describes how differentiable maps of manifolds into Euclidean spaces with singularities are related to the topological or differentiable structures of manifolds. Singularities of differentiable maps are formulated locally in principle: however, maps with certain singularities as a whole or the singularities in total carry global information. The study of differentiable maps with singularities from such kind of a viewpoint is called the global singularity theory of differentiable maps. In this chapter, we first focus on differentiable maps with only definite fold singularities, called special generic maps, and see how such maps affect the differentiable structures of the source manifolds. Then, we introduce the notion of cobordisms for maps with prescribed singularities, which will be used to extract certain invariants of singular maps and the source manifolds. We will see that singular fibers play important roles in studying such cobordisms. Finally, we give a brief exposition of a result due to Gromov, which relates the simplicial volume of a manifold with the number of certain singular fibers.