Survey on $L^2$-torsion and its (future) applications

$L^2$-Betti numbers have played and continue to play an important role in group theory, often as obstructions to some properties of groups. If all $L^2$-Betti  numbers vanish, then one can define a secondary invariant, the $L^2$-torsion, which is the $L^2$-analogue of the classical notion of Reidemeister torsion. It is a more sophisticated and richer but also harder to analyze invariant.  In this talk we want to describe some of its applications and relations to group theory and 3-manifolds which should illustrate its potential. Finally  we discuss some interesting open problems about $L^2$-torsion. The talk is meant as a survey and not as a talk conveying technical details.