Since its discovery by Barkhausen in 1919, the jerky motion of domain walls in bulk soft magnetic materials has represented a unique tool to study the microscopical processes responsible for the magnetic hysteresis. For long time, the description of this complex motion has been purely phenomenological, without any precise connection with the magnetization processes involved or the material microstructure. In the last years, using different approaches proper of mechanical statistics, new microscopical models has been introduced, offering the possibility to link the observed statistical properties of the noise to some material parameters. In particular, the Barkhausen jumps are found to exhibit universal properties, with the size and duration distributions showing extended scaling regions. Moreover, the properties of the noise of different materials can be grouped into two universality classes, depending only on the strength of long range interactions. We review all these aspects and peculiarities of the recent studies, with a particular emphasis of the still existing differences between the available experimental data and the theoretical predictions. We also show how this approach is useful to investigate the general properties of magnetic hysteresis, and the dynamics of domain walls in thin films, an important technological open problem strongly debated in the recent literature.