0. Introduction 1. Basic definitions and exact sequences 2. Universal properties 3. Duality of categories 4. Split and adjusted cotorsion modules 5. The ring $H$ of $R$-endomorphisms of $K$ 6. Completions 7. Completions of finitely generated $R$-modules 8. Localizations 9. Homological dimensions 10. $hd_RQ = 1$ 11. The role of countability