Contents. PART I: FUNCTIONAL ANALYSIS. 1. Abstract spaces. 2. Linear transformations. 3. Vector-valued functions. 4. Banach algebras. 5. Analysis in a abanch algebra. 6. Laplace integrals and binomial series. PART II: BASIC PROPERTIES OF SEMI-GROUPS. 7. Subadditive functions. 8. Semi-modules. 9. Addition theorems in a banach algebra. 10. Semigroups in the strong topology. 11. Generator and resolvent. 12. Generation of semi-groups. PART III: ADVANCED ANALYTICAL THEORY OF SEMI-GROUPS. 13. Perturbation theory. 14. Adjoint theory. 15. Operational calculus. 16. Spectral theory. 17. Holomorphic semi-groups. 18. Applications to ergodic theory. PART IV: SPECIAL SEMI-GROUPS AND APPLICATIONS. 19. Translation and powers. 20. Trigonometric semi-groups. 21. Semi-groups in Lp. 22. Semi-groups in Hilbert space. 23. Miscellaneous applications. PART V: EXTENSION OF THE THEORY. 24. Notes on Banach algebras. 25. Lie semi-groups. 16. Functions on vectors to vectors.