With this survey on the Hermite transformation we want to pursue the following two goals. First, we want to provide a comprehensive and up-to-date description of the Hermite transformation, its underlying philosophy, and its most important properties and their implications for applications. As so often when publications and development go hand-in-hand, new insights have led to changes in or generalizations of already published results, and not all of these changes have been considered sufficiently substantial to be published separately. As a consequence, the existing publications on the Hermite transformation do not fully reflect our most recent insights, and the current paper intends to remedy this. Second, we also want to share some new results. Two specific new results, that is, partial signal decompositions and intersection curvatures, are therefore treated in more detail than other aspects.