In this chapter, we study the theory of one-dimensional Fourier transforms, the inversion formula, convergence and summability of Fourier transforms. In the first two sections, we introduce the Fourier transform for Schwartz functions and we extend it to L2(ℝ), L1(ℝ), Lp(ℝ)(1≤p≤2) functions as well as to tempered distributions. We prove some elementary properties and the inversion formula. In Sect. 2.4, we deal with the convergence of Dirichlet integrals. Using some results for the partial sums of Fourier series proved in Sect. 2.3, we show that the Dirichlet integrals converge in the Lp(ℝ) -norm to the function (1 < p < ∞). The proof of Carleson’s theorem, i.e. that of the almost everywhere convergence can be found in Carleson , Grafakos , Arias de Reyna , Muscalu and Schlag , Lacey  or Demeter .