Fourier Series are a powerful tool in Applied Mathematics; indeed, their importance is twofold since Fourier Series are used to represent both periodic real functions as well as solutions admitted by linear partial differential equations with assigned initial and boundary conditions. To idea inspiring the introduction of Fourier Series is to approxi- mate a regular periodic function, of period T , via a linear superposition of trigonometric functions of period the same T: thus, Fourier Polynomials are constructed. They play, in the case of regular periodic real functions, a role analogue to that one of Taylor Polynomials when smooth real functions are considered. An overview on the theory of Fourier Series is provided together with some explicit examples.