The Naming Game is a model of non-equilibrium dynamics for the self-organized emergence of a linguistic convention or a communication system in a population of agents with pairwise local interactions. We present an extensive study of its dynamics on complex networks, that can be considered as the most natural topological embedding for agents involved in language games and opinion dynamics. Except for some community structured networks on which metastable phases can be observed, agents playing the Naming Game always manage to reach a global consensus. This convergence is obtained after a time generically scaling with the populations size $N$ as $t_{conv} sim N^{1.4 pm 0.1}$, i.e. much faster than for agents embedded on regular lattices. Moreover, the memory capacity required by the system scales only linearly with its size. Particular attention is given to heterogenous networks, in which the dynamical activity pattern of a node depends on its degree. High degree nodes have a fundamental role, but require larger memory capacity. They govern the dynamics acting as spreaders of (linguistic) conventions. The effects of other properties, such as the average degree and the clustering, are also discussed.
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