The construction and classification of Riemannian metrics satisfying certain curvature bounds are of fundamental interest in differential geometry. In this talk we specialize to metrics of positive scalar curvature, which are characterized by a simple volume growth condition of balls with small radii. Research during the last couple of years, based on a variety of methods, revealed the topological complexity of the space of all such metrics on a fixed manifold. We will give an overview of this development.