I. Introduction. - I. 1. Preliminaries from algebra and analysis. - I. 2. Preliminaries from automata and formal language theory. - I. 3. Formal power series in noncommuting variables. - II. Rational series. - II. 1. Rational series and linear systems. - II. 2. Recognizable series. - II. 3. Hankel matrices. - II. 4. Operations preserving rationality. - II. 5. Regular languages and rational series. - II. 6. Fatou properties. - II. 7. On rational series with real coefficients. - II. 8. On positive series. - II. 9. Rational sequences. - II. 10. Positive sequences. - II. 11. On series in product monoids. - II. 12. Decidability questions. - III. Applications of rational series. - III. 1. On rational transductions. - III. 2. Families of rational languages. - III. 3. Rational series and stochastic automata. - III. 4. On stochastic languages. - III. 5. On one-letter stochastic languages. - III. 6. Densities of regular languages. - III. 7. Growth functions of L systems: characterization results. - III. 8. Growth functions of L systems: decidability. - IV. Algebraic series and context-free languages. - IV. 1. Proper algebraic systems of equations. - IV. 2. Reduction theorems. - IV. 3. Closure properties. - IV. 4. Theorems of Shamir and Chomsky-Schiitzenberger. - IV. 5. Commuting variables and decidability. - IV. 6. Generalizations of proper systems. Fatou extensions. - IV. 7. Algebraic transductions. - Historical and bibliographical remarks. - References. Language: English