The theories we are interested in have many different models: a natural way to relate them is to consider the maps between them that preserve first-order information. This chapter is entirely devoted to such maps. In Sect. 4.1 we focus on maps that preserve atomic formulae. In Sect. 4.2 we restrict attention to the elementary maps, which transfer all first-order information back and forth between structures, and define the notion of substructure and elementary substructure in terms of maps. In Sect. 4.3 we study an important connection between the existence of maps relating structures and the satisfiability of sentences, using the fundamental notions of a diagram and an elementary diagram. In Sect. 4.4 we offer a philosophical application of some model-theoretic ideas discussed in this chapter.