Periodic Functions. Trigonometric Systems. Modes of Convergence. Complete Orthogonal Systems. The Riemann-Lebesgue Lemma. The Dirichlet Kernel. Piecewise Smooth Functions. Gibb's Phenomenon and the Localization Principle. Trigonometric Series. Approximation by Trigonometric Functions. Parseval's Theorem. Sums and Products of Fourier Series. Integration and Differentiation of Fourier Series. Convergence Improvement. Cesaro Sums and the Fejer Kernel. Abel Sums and the Poisson Kernel.