The Cat and the Butterfly: quantum properties of chaotic systems

In this article I present an introduction to the problem of quantum chaos, i.e. the characterization and treatment of quantum systems whose classical analogue has chaotic dynamics. Two of the main approaches to this problem are discussed: one is statistical, making use of random matrices, while the other is based on the semiclassical approximation. Results obtained by these approaches are mentioned for closed systems (spectral correlation functions), transport (moments like conductance and shot noise) and resonances (fractal Weyl law and fractal wave functions).