Kurzbeschreibung: The classical Minkowski problem asks for necessary and sufficient con- ditions such that a finite Borel measure on the sphere is the surface area measure of a convex body. For a discrete measure this becomes the problem to decide under which conditions a convex polytope with prescribed normal directions and prescribed area of its facets exists. This problem was solved by Minkowski and it is a corner stone of clas- sical Brunn-Minkowski theory. The analogue problem in modern convex geometry and within the Lp-Brunn-Minkowski-theory is known as the Lp- Minkowski problem. Of particular interest is the limit case p = 0 and the associated logarithmic Minkowski problem which asks for a characterization of the so called cone volume measure of a convex body. In the discrete setting this leads to the problem of deciding when a convex polytope with prescribed normal directions and prescribed volumes of the cones generated by the origin and its facets exists. In the talk we survey on the current state of the art of the logarithmic Minkowski problem.