I'll survey some results and open problems about changes of cofinality when we go from the universe of sets V to some outer model W. There are some surprising constraints, for example if κ is regular in V and (κ⁺)^V is still a cardinal in W while κ collapses, then cf(κ) = cf(|κ|) in W. One purpose of the talk is to provide some background for the set theory reading group.