We have already introduced quadratic number ﬁelds and their discriminants in Chapter 1 (see Deﬁnition 1.1.8 and Theorem 1.1.9). In this chapter we introduce and investigate lattices and orders in quadratic number ﬁelds as well as ideals and ideal classes. These are the basic algebraic objects built by quadratic irrationals. Their structure will be the basis for a deeper understanding of Gauss’ theory of binary quadratic forms in Chapter 6. Apart from that, the algebraic theory of quadratic orders is of independent interest, and we shall investigate it in detail.