This chapter presents basic results on pointwise convergence of Fourier series. As a fundamental example, the Fourier series of the sawtooth function is studied. Properties of this series are deduced, such as pointwise convergence and uniform convergence in closed intervals, that do not contain a discontinuity point. The Gibbs phenomenon is worked out. The theorems of Dirichlet and Fejér are presented and discussed with examples. Their rigorous proofs are postponed until Chap. 7 . Examples and exercises help the reader become familiar with the necessary calculations.