This report discusses the question whether or not a given unstable module is the mod-$p$ cohomology of a space. One first discusses shortly the Hopf invariant 1 problem and the Kervaire invariant 1 problem and gives their relations to homotopy theory and geometric topology. Then one describes some more qualitative results, emphasizing the use of the space map$(B\mathbb Z/p, X)$ and of the structure of the category of unstable modules.