Mathematical models of biochemical reaction networks give rise to deterministic or stochastic dynamical systems that are usually high dimensional, nonlinear, and have many unknown parameters. Nevertheless, it is often possible to draw strong conclusions on the dynamics of such systems based on graph-theoretical properties of the reaction network. Moreover, we show that these results can be generalized to yield criteria for global injectivity for large classes of nonlinear maps. We also explain how these results relate to other problems, such as the Jacobian Conjecture in algebraic geometry and Bezier self-intersection in computer graphics