About 30 years ago, Rick Schoen asked whether a properly embedded (nonzero) constant mean curvature (CMC) surface in R^3 with finite total absolute curvature must be the sphere or the cylinder.  By an equally old result of the speaker (with Nick Korevaar and Bruce Solomon), this question is equivalent to asking if all its ends are asymptotic to a cylinder.  After reviewing necksize bounds (suggesting the answer is "yes") and some experiments (suggesting the answer is "no"), we'll discuss our ongoing work (with Jeremy Leach and Rafe Mazzeo) that constructs such CMC surfaces, as well as our analogous construction for complete metrics with (positive) constant scalar curvature (CSC) on locally conformally flat manifolds (of dimension at least 3).