Two classical theorems for orders in semisimple \({\mathbb{Q}}\)-algebras are extended to the non-semisimple case: 1. the existence of maximal orders, and 2. the Jordan-Zassenhaus theorem. Ad 1: An order \(\Lambda\) in a \({\mathbb{Q}}\)-algebra A is said to be maximal if for any overorder \(\Gamma\) \(\supset \Lambda\) there exists an algebra-automorphism \(\alpha\) with \(\alpha\) \(\Gamma\) \(\subset \Lambda\). It is shown that maximal orders exist in A if either A is representation-finite or if \(Rad^ 3A=0\). Ad 2: For an order \(\Lambda\) in A, two \(\Lambda\)-ideals I, J are said to belong to the same exact ideal class if they are isomorphic as left \(\Lambda\)-modules and have \(\Lambda\) as left multiplier \(\Lambda =O_{\ell}(I)=O_{\ell}(J)\). The number of exact ideal classes is shown to be finite. For two special classes of orders (1. those which correspond to integral quadratic forms and 2. orders in \({\mathbb{Q}}[x]/(x^ n)\) occurring in the similarity problem for nilpotent integral matrices) the Frobenius duality of A is used to obtain an embedding of the exact ideal classes into a finite lattice with duality.