This is a survey. We study the question to what extend classical Hodge-deRham theory for harmonic differential forms carries over to harmonic spinors. Despite some special phenomena in very low dimensions and despite the Atiyah-Singer index theorem which provides a link between harmonic spinors and the topology of the underlying manifold it turns out that in many dimensions harmonic spinors are not topologically obstructed. In this respect harmonic spinors behave very differently from harmonic differential forms. We also discuss parallel spinors and Killing spinors