We study mesoscopic features of classically integrable systems, in particular, spectral correlations and orbital magnetism in a rectangular box. Such systems are easy to describe in terms of individual energy levels and wave functions of non-interacting particles. However, in the semiclassical regime, when the large number of levels come into play and large numbers of particles are responsible for the response to external perturbations, the problem becomes essentially strongly interacting as manifested by very long correlations in the energy space.

The central result of this Thesis is the exact analytical description, as well as numerical verification, of such correlations that result in remarkable non-decaying oscillations of the level number variance in an energy interval as a function of the interval width. Such predictable functional behavior of a statistical quantity had not been previously realized. While describing the property of the (semiclassical) quantum spectrum, the physics behind this phenomenon is due to a few shortest classically periodic orbits in the system.

We also predict that such systems are characterized by extreme non-self averaging of physical properties, such as orbital susceptibility of free electrons. It means that even the average, but also higher moments, are ill-defined in the absence of cut-offs, such as due to thermal averaging and inelastic collisions. the most striking conclusion of this research is that, despite appearance of simplicity relative to classically chaotic, disordered systems that received a lion's share of attention in mesoscopic and nano physics, the classically integrable systems are much more complex both in terms of their analytical description and physical properties that they should exhibit experimentally.