The Hurwitz existence problem for surface branched covers

To a branched cover f:Σ ˜→Σ between closed surfaces one can associate combinatorial datum given by the topological types of Σ ˜ and Σ, degree d f, number n branching points partitions local degrees f at preimages points. This must satisfy Riemann-Hurwitz condition plus some extra ones if either or both are non-orientable. A very old question posed Hurwitz [14] in 1891 asks whether satisfying these necessary conditions is actually realizable (namely, associated to existing f) not (in which case it called exceptional). Or, more generally, count realizations up natural equivalence relation. Many partial answers have been problem over time, but complete solution still missing. In this short course we will report on ancient recent results techniques employed attack question.