I. Fourier transforms on L1 (-?,?).- Basic properties and examples.- The L1 -algebra.- Differentiability properties.- Localization, Mellin transforms.- Fourier series and Poisson's summation formula.- The uniqueness theorem.- Pointwise summability.- The inversion formula.- Summability in the L1-norm.- . The central limit theorem.- . Analytic functions of Fourier transforms.- . The closure of translations.- . A general tauberian theorem.- . Two differential equations.- . Several variables.- II. Fourier transforms on L2(-?,?).- Introduction.- Plancherel's theorem.- Convergence and summability.- The closure of translations.- Heisenberg's inequality.- Hardy's theorem.- The theorem of Paley and Wiener.- Fourier series in L2(a,b).- Hardy's interpolation formula.- . Two inequalities of S. Bernstein.- . Several variables.- III. Fourier-Stieltjes transforms (one variable).- Basic properties.- Distribution functions, and characteristic functions.- Positive-definite functions.- A uniqueness theorem.- Notes.- References.