Seven short stories on blowups and resolutions

The lectures adress to students and geometers who are not experts in the field, but who need to use blowups occasionally or who just want to have a good comprehension of them. References are scattered in the literature and mostly concentrate on only part of the story. This text is neither complete, but hints at least at the variety of properties, results and techniques which are related to blowups and which make them so attractive. Actually, it may serve as the starting point to write a comprehensive treatise on blowups (which should in particular include the solutions to all exercises). The obvious objection from algebraic geometers to such a project will be that blowups are too simple to deserve a separate treatment. The many open and intricate questions listed in these notes may serve as a reply to this reproach.