The classical notions of structural controllability and structural observability are receiving increasing attention in Network Science, as they provide a mathematical basis to understand how the network structure of a dynamic system affects its controllability and observability properties. However, these two notions are formulated assuming systems with linear dynamics, which significantly limits their applicability. To overcome this limitation, here we introduce and fully characterize the notions of “structural accessibility” and “structural observability” for the broad class of nonlinear systems with meromorphic dynamics.We show how nonlinearities render easier the problem of controlling and observing networked systems, in the sense that meromorphic nonlinearities reduce the number of variables that are necessary to directly control and directly measure. Our results contribute to clarify the role that the network structure and the nonlinearities play in our ability to control and observe complex dynamic systems.