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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.