This overview is based on the talk 101] given at the mini-workshop 0648c Dirac operators in di?erential and non-commutative geometry, Mat- matisches Forschungsinstitut Oberwolfach. Intended for non-specialists, it explores the spectrum of the fundamental Dirac operator on Riemannian spin manifolds, including recent research and open problems. No background in spin geometry is required; nevertheless the reader is assumed to be fam- iar with basic notions of di?erential geometry (manifolds, Lie groups, vector and principal bundles, coverings, connections, and di?erential forms). The surveys ...