A new coarse-grained representation of the dynamics of neuronal networks is introduced and developed in terms of kinetic equations, which, via a novel moment closure, are derived mathematically, directly from the original large-scale integrate-and-fire (I&F) network. This powerful kinetic theory can capture the full dynamic range of neuronal networks ---from the mean-driven limit (a limit as the number of neurons N??, in which the fluctuations vanish) to the fluctuation-dominated limit (such as in small N networks). Comparison with full numerical simulations of the original I&F network establishes that the reduced dynamics is very accurate and numerically efficient over all dynamic ranges. Both analytical insights and numerical scale-up of neuronal computation can be achieved via this kinetic approach. Here the theory is illustrated by a study of the dynamical properties of networks of various architectures, including excitatory and inhibitory neurons of both simple and complex type, which exhibit rich phenomena, e.g., transition to bistability and hysteresis, even in the presence of large fluctuations. The implication for possible connections between the structure of the bifurcations and the behavior of complex cells is discussed. Finally, I&F networks and kinetic theory are used to discuss orientation selectivity of complex cells for "ring model" architectures which characterize changes in the response of neurons located from near to far from "orientation pinwheel centers".