Survey on aspherical manifolds

We give a survey over aspherical closed manifolds. Aspherical is the purely homotopy theoretic condition that the universal covering is contractible. There are many well-known examples such as closed Riemannian manifolds with non-positive sectional curvature, but also very exotic examples such as closed aspherical manifolds that do not admit a triangulation. We discuss some prominent conjectures, e.g., the Borel Conjecture about the topological rigidity, and the Singer Conjecture about the concentration of L^2-Betti numbers in the middle dimension. The main point is to get a better understanding why the condition aspherical has so many consequences about the topology, geometry, algebra, and analysis about such manifolds.