The cellular uplink has typically been studied using simple Wyner-type analytical models where interference is modeled as a constant or a single random variable, or via complex system-level simulations for a given set of parameters, which are often insufficient to evaluate performance in all operational regimes. In this paper, we take a fresh look at this classic problem using tools from point process theory and stochastic geometry, and develop a new tractable model for the cellular uplink which provides easy-to-evaluate expressions for important performance metrics such as coverage probability. The main idea is to model the locations of mobiles as a realization of a Poisson Point Process where each base station (BS) is located uniformly in the Voronoi cell of the mobile it serves, thereby capturing the dependence in two spatial processes. In addition to modeling interference accurately, it provides a natural way to model per-mobile power control, which is an important aspect of the uplink and one of the reasons why uplink analysis is more involved than its downlink counterpart. We also show that the same framework can be used to study regular as well as irregular BS deployments by choosing an appropriate distribution for the distance of a mobile to its serving BS. We verify the accuracy of this framework with an actual urban/suburban cellular network.