Mathematical diffraction theory is concerned with the determination of the diffraction image of a given structure and with the corresponding inverse problem of structure determination. The understanding of systems with continuous and mixed spectra has improved considerably in recent years. The phenomenon of homometry shows various unexpected new facets, in particular when considering systems with stochastic components. After a brief introduction and a summary of pure point spectra, classic deterministic examples with singular or absolutely continuous spectra are discussed. In particular, an isospectral family of structures with continuously varying entropy is presented.