Given a group W and a set S of involutory generators, when does \((W,S)\) deserve to be called an “abstract reflection group” (or a “Coxeter system”)? In this chapter we give the two answers alluded to in Sect. 1.1 . The first is that for each \(s\in S\) its fixed set separates \(\operatorname {Cay}(W,S)\) (see Sect. 3.2). The second is that W has a presentation of a certain form (see Sect. 3.3). The main result, Theorem 3.3.4, asserts these answers are equivalent. Along the way we find three combinatorial conditions (D), (E) and (F) on \((W,S)\) , each of which is equivalent to it being a Coxeter system. This line of reasoning culminates with Tits’ solution of the word problem for Coxeter groups, which we explain in Sect. 3.4.