  This is a survey on known results and open problems about closed aspherical
manifolds, i.e., connected closed manifolds whose universal coverings are
contractible. Many examples come from certain kinds of non-positive curvature
conditions. The property aspherical which is a purely homotopy theoretical
condition implies many striking results about the geometry and analysis of the
manifold or its universal covering, and the ring theoretic properties and the
K- and L-theory of the group ring associated to its fundamental group. The
Borel Conjecture predicts that closed aspherical manifolds are topologically
rigid. The article contains new results about product decompositions of closed
aspherical manifolds and an announcement of a result joint with Arthur Bartels
and Shmuel Weinberger about hyperbolic groups with spheres of dimension greater
or equal to six as boundary. At the end we describe (winking) our universe of
closed manifolds.
