Seshadri constants express the so called local positivity of a line bundle on a projective variety. They were introduced in [Dem92] by Demailly. The original hope of using them towards a proof of the Fujita conjecture was too optimistic, but it soon became clear that they are interesting invariants quite in their own right. Lazarsfeld's book [PAG] contains a whole chapter devoted to local positivity and serves as a very enjoyable introduction to Seshadri constants. Since this book has appeared, the subject witnessed quite a bit of development. It is the aim of these notes to give an account of recent progress as well as to discuss many open questions and provide some examples. The idea of writing these notes occurred during the workshop on Seshadri constants held in Essen 12-15 February 2008.