In this thesis work is described that arose out of a study of harmonic Riemannian manifolds. A definition of harmonicity is given and from this it is shown how the Ledger conditions on the curvature of a harmonic manifold may be derived in principle and the first four are written down. The first three Ledger conditions are put into local co-ordinate form and simpler conditions are derived, the most important being the super-Einstein condition. The idea of the Schur property is also introduced. The mean-value work of Gray and Willmore is described and extended as far as the r(^8) term under some simplifying conditions. Finally there is an investigation of the extent to which the compact classical simple Lie groups with bi-invariant metrics can satisfy Ledger’s first three conditions.