I. Fourier transforms on L1 (-?,?) -- ยง1. Basic properties and examples -- ยง2. The L1 -algebra -- ยง3. Differentiability properties -- ยง4. Localization, Mellin transforms -- ยง5. Fourier series and Poissonโ{128}{153}s summation formula -- ยง6. The uniqueness theorem -- ยง7. Pointwise summability -- ยง8. The inversion formula -- ยง9. Summability in the L1-norm -- ยง10. The central limit theorem -- ยง11. Analytic functions of Fourier transforms -- ยง12. The closure of translations -- ยง13. A general tauberian theorem -- ยง14. Two differential equations -- ยง15. Several variables -- II. Fourier transforms on L2(-?,?) -- ยง1. Introduction -- ยง2. Plancherelโ{128}{153}s theorem -- ยง3. Convergence and summability -- ยง4. The closure of translations -- ยง5. Heisenbergโ{128}{153}s inequality -- ยง6. Hardyโ{128}{153}s theorem -- ยง7. The theorem of Paley and Wiener -- ยง8. Fourier series in L2(a,b) -- ยง9. Hardyโ{128}{153}s interpolation formula -- ยง10. Two inequalities of S. Bernstein -- ยง11. Several variables -- III. Fourier-Stieltjes transforms (one variable) -- ยง1. Basic properties -- ยง2. Distribution functions, and characteristic functions -- ยง3. Positive-definite functions -- ยง4. A uniqueness theorem -- Notes -- References