In this chapter we introduce the central idea of this book—Serre’s notion of G-complete reducibility for subgroups of a reductive group G—and set up some of the questions which motivate us for the rest of the book. In particular, we give a first discussion of the ideas of ascent and descent of complete reducibility from and to a reductive subgroup M of G, Questions 4.1.8 and 4.1.9. We also collect some of the basic properties of completely reducible subgroups, including the important fact that a G-completely reducible subgroup of G is reductive (Lemma 4.1.6) and record some aspects of the behaviour of G-complete reducibility in the presence of an algebraically closed extension field.