This chapter undergoes a systematic study of the fundamental concepts of Barabanov norm and extremal trajectories for arbitrary switchings. Necessary and sufficient conditions are provided concerning their existence, and several properties are established, for instance the characterization of the extremal trajectories via optimal control. Applications of Barabanov norms are given, namely proving a Berger-Wang formula and the continuity of the growth rate, and also investigating the concept of resonance for switched linear systems.