The main subject of the thesis is the study of Fermi acceleration, regarded nowadays as a fundamental acceleration mechanism, consisting in the increase of the mean energy of particles due to collisions with moving scatterers. Prior to the study of extended systems, the prototype one-dimensional dynamical system exhibiting Fermi acceleration was considered; the stochastic Fermi-Ulam model. The analysis of the dynamics in this system revealed the pitfalls of the standard, widely used, quasi-static approximation, which neglects the impact of the location of the collision events in the configuration space. In order to take this into account, a novel approximation scheme was introduced allowing both the analytical treatment of the acceleration process as well as fast numerical simulations. Furthermore, the limitations and possible inconsistencies stemming from the treatment of Fermi acceleration via the Fokker-Planck equation were brought to light. A new self-consistent methodology on the basis of the Chapman-Kolmogorov equation was put forward, capable of giving an accurate description of Fermi acceleration for all times. The understanding gained through the investigation of the Fermi-Ulam model, was then utilized for the study of Fermi acceleration in spatially extended systems, using as a prototype the two-dimensional randomized Lorentz gas. The newly introduced approximation was generalized for application to higher dimensional systems. The study revealed that the increase of the efficiency of Fermi acceleration depends only on the symmetries of the driving time-law and is insensitive to the geometrical properties of the moving scatterers and the dimensionality of the time-dependent system. Finally, the study of the driven Lorentz gas, in a channel geometry, revealed a completely new aspect of Fermi acceleration, linking it, for the first time, to the phenomenon of self-organized criticality. Particularly it was shown that Fermi acceleration permits the spontaneous synchronization of the motion of the particles with that of the moving scatterers, such that particles can travel collision-free for long times. This, in turn, gives rise to strongly intermittent dynamics, which, as it was shown, is a sufficient condition for the emergence of scale-free cross-correlations between non-interacting particles.
