This thesis presents an investigation of the models describing electrical conductivity in polymer nanocomposites, which consist in more or less random dispersions of nanometric conductive fillers like carbon nanotubes, nanofibers or graphene sheets in a polymer matrix. The investigation is carried out mainly through simulations with ad-hoc developed algorithms coupled with analytical studies of both the global system connectivity and the details of a realistic electron tunneling inter-particle conduction mechanism. Conductive polymer nanocomposites manifest a sudden increase of the bulk conductivity when the content of the conductive species exceeds a certain threshold, a behavior shared with many other insulator-conductor biphasic systems. Percolation theory remains the most widely used theoretical formulation to describe for such transition. Its embodiment which most faithfully takes into account the microscopic features of the composite is the hard-core-penetrable-shell model, and in the present work has been extended to a wide range of anisotropic filler particle shapes modeled as hard ellipsoids-of-revolution. However, such connectivity-based description of the insulator transition manifests some fundamental incompatibilities when applied to the specific case of nanocomposites. Indeed, the percolation formulation requires the connections between the filler particles to be of the "on-off" binary sort, but this is in contrast with a microscopically justified electron tunneling inter-particle conduction mechanism, which entails no sharp cutoff of the connectivity. Since the tunneling conductance decays exponentially over the distance with a characteristic decay length in the order of a few nanometers, for macroscopic fillers an abrupt cutoff description of inter-particle connectivity may still be suitable. This ceases to be valid for nanometric fillers, which have one or more characteristic dimensions that are comparable with the distances of tunneling. The solution of the tunneling-percolation problem in the nanocomposite regime is the main focus of this thesis. We present a model of conductivity where the filler particles form a network of globally connected objects via tunneling. Such a model does not need any abrupt interruption to induce the insulator-conductor transition and is able to reproduce the typical conductivity versus nanofiller volume fraction curves found experimentally. In this description, the transition is interpreted as the crossover region where the conductivity contribution due to the tunneling network overtakes the intrinsic one of the polymer matrix. Furthermore, we introduce an approximation route and explicit formulas based on the critical path method, which allow a quick and precise estimation of the overall system conductivity for various commonly employed nanocomposites. The validity of our formulation is verified by extracting from a large number of experimental results the characteristic tunneling length, which is found to be within the expected range of its values. Finally, we consider the case of graphite polymer composites in more detail, and we introduce a simple model of conductivity, which is able to account for some of the features of these materials.