The title of this chapter seems to suggest a plunge into the depths of abstraction, populated with great hypotheses, axioms, lemmas, theorems, proofs and other mathematical arcana. This is not the case, and the formalities here, such as they are, are friendly. The theory of transformations does not have hard boundaries, recognized and certified by those so disposed to do these things. The material presented here, however, does lay an adequate and important foundation for what follows. Here you will meet functions, mappings and transformations key players in this drama. Linear transformations and geometric invariants immediately follow and round out the preliminaries. The final sections introduce isometries, similarities, affinities, projectivities, and topological transformations. Here the text describes fundamental concepts, usually informally and without recourse to a more rigorous axiomatic approach. For the isometries, et al, the mode is at first synthetic and diagrammatic, to foster an intuitive understanding. The mode then shifts to the discussion of classical Cartesian or analytic equations describing each of these transformations.