Nanofillers with highly anisotropic shapes, such as carbon nanotubes, graphene nanoplatelets, carbon black, and metallic nanowires are used as inclusions in polymer matrix materials to generate nanocomposites with superior electrical, mechanical, and thermal properties. In this paper, we report on our recent and ongoing studies focusing on the enhanced effective electrical conductivity of such composites. First, we report on Monte Carlo simulations of systems of polydisperse prolate and oblate ellipsoids using the critical path-based tunneling-percolation mode. For polydisperse prolate ellipsoids, the critical percolation volume fraction, fc, is shown to have a quasi-universal dependence on the weight-averaged aspect ratio. For polydisperse oblate ellipsoids, fc is shown to have a quasi-universal dependence on the apparent aspect ratio, which is a function of up to fourth moment of the size distribution, as given by percolation theory. In both cases, the function approaches the theoretical predictions for higher volume fractions and higher aspect ratios. The model predictions are then compared with experimental data to estimate the tunneling length scale which is found to be within the expected range. Next, we examine the effect of filler alignment on percolation behavior of nanocomposites using Monte Carlo simulations of monodisperse prolate and oblate hard-core soft-shell ellipsoids representing carbon nanotubes and graphene nanoplatelets, respectively. As expected, the percolation threshold is observed to increase with increasing extent of alignment. For a highly aligned system of rod-like fillers, the simulation results are shown to be in good agreement with the second virial approximation-based predictions. However, for a highly aligned system of disk-like fillers, the second virial approximation-based results are observed to significantly deviate from the simulations, even for higher aspect ratios. The anisotropy in percolation threshold is found to vanish with increasing system size even for highly aligned systems of fillers.