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Our ability to manipulate the behavior of complex networks depends on the design of efficient control algorithms and, critically, on the availability of an accurate and tractable model of the network dynamics. While the design of control algorithms for network systems has seen notable advances in the past few years, knowledge of the network dynamics is a ubiquitous assumption that is difficult to satisfy in practice, especially when the network topology is large and, possibly, time-varying. In this paper we overcome this limitation, and develop a data-driven framework to control a complex dynamical network optimally and without requiring any knowledge of the network dynamics. Our optimal controls are constructed using a finite set of experimental data, where the unknown complex network is stimulated with arbitrary and possibly random inputs. In addition to optimality, we show that our data-driven formulas enjoy favorable computational and numerical properties even compared to their model-based counterpart. Although our controls are provably correct for networks with linear dynamics, we also characterize their performance against noisy experimental data and in the presence of nonlinear dynamics, as they arise when mitigating cascading failures in power-grid networks and when manipulating neural activity in brain networks.