The Isomorphism Conjectures due to Paul Baum and Alain Connes and to Tom Farrell and Lowell Jones are important conjectures, which have many interesting applications and consequences. However, they are not easy to formulate and it is a priori not clear why the actual versions are the most promising ones. The current versions are the final upshot of a longer process, which has led to them step by step. They have been influenced and steered by various new results that have been proved during the last decades and given new insight into the objects, problems, and constructions at which these conjectures aim.