Fidelity serves as a benchmark for the relieability in quantum information processes, and has recently atracted much interest as a measure of the susceptibility of dynamics to perturbations. A rich variety of regimes for fidelity decay have emerged. The purpose of the present review is to describe these regimes, to give the theory that supports them, and to show some important applications and experiments. While we mention several approaches we use time correlation functions as a backbone for the discussion. Vanicek's uniform approach to semiclassics and random matrix theory provides an important alternative or complementary aspects. Other methods will be mentioned as we go along. Recent experiments in micro-wave cavities and in elastodynamic systems as well as suggestions for experiments in quantum optics shall be discussed.Comment: Review article with some original results in integrable systems and random matrix models; 133 pages, 53 figure

We study statistical properties of the ensemble of large NϫN random matrices whose entries H i j decrease in a power-law fashion H i j ϳ͉iϪ j͉ Ϫ␣ . Mapping the problem onto a nonlinear model with nonlocal interaction, we find a transition from localized to extended states at ␣ϭ1. At this critical value of ␣ the system exhibits multifractality and spectral statistics intermediate between the Wigner-Dyson and Poisson statistics. These features are reminiscent of those typical of the mobility edge of disordered conductors. We find a continuous set of critical theories at ␣ϭ1, parametrized by the value of the coupling constant of the model. At ␣Ͼ1 all states are expected to be localized with integrable power-law tails. At the same time, for 1Ͻ␣Ͻ3/2 the wave packet spreading at a short time scale is superdiffusive: ͉͗r͉͘ϳt 1/(2␣Ϫ1) , which leads to a modification of the Altshuler-Shklovskii behavior of the spectral correlation function. At 1/2Ͻ␣Ͻ1 the statistical properties of eigenstates are similar to those in a metallic sample in dϭ(␣Ϫ1/2) Ϫ1 dimensions. Finally, the region ␣Ͻ1/2 is equivalent to the corresponding Gaussian ensemble of random matrices (␣ϭ0). The theoretical predictions are compared with results of numerical simulations.