This thesis is devoted to the study of phenomenological consequences of theoretical models of Quantum Gravity. In particular, this work is focused on the study of possible violations of Lorentz invariance, which may arise if, owing to quantum gravity effects, the high-energy structure of the spacetime is different from the smooth, continuous one we are used to in our low-energy world. After a brief description of the most widely known models accounting for Lorentz invariance violations, particular focus will be given to astrophysical tests of Lorentz invariance. These are motivated by the fact that some astrophysical objects are able to accelerate particles to extremely high energies, unreachable to terrestrial experiments. This consideration naturally leads us to look at the radiation of the Crab Nebula, one of the most powerful objects in our Galaxy. We first understand how the violation of Lorentz invariance affects the physical processes at the basis of the production of electromagnetic radiation by this object. Then, we compare our prediction for the Lorentz violating spectrum to observational data, exploiting the vast multi-wavelength information on the Crab Nebula radiation. Furthermore, we take advantage of the recent development of new technology to improve on our analysis of the Crab Nebula radiation by extending our research to the effects of Lorentz violation onto hard X-ray polarization. After this investigation we shall move to study the physics of cosmic rays, the most energetic particles ever experienced on Earth. Our interest in this physics is twofold: on the one hand, we want to understand more about their properties and their propagation. To this aim, we develop a new model of propagation for cosmic rays in our Galaxy, exploiting as much as possible of the multi-channel information available at present. On the other hand, according to the multi-channel perspective, we try to understand the consequences of Lorentz symmetry violation on the properties of ultra-highenergy cosmic rays.
