Boolean functions (which transform a finite sequence of 0/1 bits, to a 0/1 value) are one of the basic objects of theoretical computer science. They arise in various areas of mathematics, as e.g. combinatorics, statistical physics or metric geometry. Their analysis has experienced a tremendous development in the last decades. In the first part of the lecture, we will give an overview of the main notions and techniques (Fourier spectrum, influences of variables, hypercontractive estimates,...) and advertise some applications of this theory, for instance to voting systems. The second part will present some results of a work in progress with P. Ivanisvilii, A. Malliaris and X. Xie, which focusses on the structure of the Fourier spectrum of low degree Boolean functions.