The previous chapter was devoted to the analysis of linear actions of Lie groups on functions, provided by either ordinary or multiplier representations. Although of great importance, such actions are certainly not the most general one can imagine, and we will have occasion to make use of fully nonlinear group actions. Such general transformation groups figure prominently in Lie's theory of symmetry groups of differential equations, which we discuss in Chapter 6. They also reoccur in the Cart an equivalence method, to be discussed in the second part of the book. The transformation groups will act on the basic space coordinatized by the independent and dependent variables relevant to the system of differential equations under consideration, and we begin with a short discussion of the different types of transformation groups of interest. Since we are dealing with differential equations we must be able to handle the derivatives of the dependent variables on the same footing as the independent and dependent variables themselves. This chapter is devoted to a detailed study of the proper geometric context for these purposes —the so-called “jet spaces” or “jet bundles”, well known to nineteenth century practitioners, but first formally defined by Ehresmann, [66]. After presenting a simplified version of the basic construction, we then discuss how group transformations are “prolonged” so that the derivative coordinates are appropriately acted upon, and, in the case of infinitesimal generators, deduce the explicit prolongation formula. Next we introduce the underlying contact structure on jet space, and present Lie's theory of contact transformations which, in the case of a single dependent variable, enlarge our repertoire of available group actions.