§ 1. Derivations.- § 2. Differential Algebras.- § 3. Universal Extension of a Differential Algebra.- § 4. Description of the Universal Extension in Special Cases.- § 5. Differential Modules of Field Extensions.- § 6. Differential Modules of Local Rings.- § 7. Differential Modules of Affine Algebras.- § 8. Smooth Algebras.- § 9. Differential Modules of Complete Intersections.- § 10. The Kahler Differents (Jacobian Ideals) of an Algebra.- § 11. Universally Finite Differential Algebras.- § 12. Differential Algebras and Completion.- § 13. Differential Modules of Semianalytic Algebras.- § 14. Regularity Criteria for Semianalytic Algebras.- § 15. Existence of p-Bases.- § 16. Traces of Differential Forms.- § 17. Residues in Algebraic Function Fields of one Variable.- Appendices.- A. Commutative Algebras.- B. Dimension Formulas in Algebras of Finite Type.- C. Complete Intersections.- D. The Fitting Ideals of a Module.- E. The Dual of a Module over a Noetherian Ring.- F. Traces.- G. Differents.- Symbol Index.