In the last years, complex-network analysis has been applied to several fields in economics, giving rise to a wide literature, both empirical and theoretical. In this talk, I will overview some recent work exploring the properties of the international-trade network (ITN), defined as the graph where nodes are world countries and links represent bilateral trade flows (imports or exports). I address five main questions: (1) Why characterizing trade flows using a network representation may be relevant for trade economists? (2) Can the knowledge of the ITN topological properties shed new light on issues like growth, globalization and trade integration? (3) Can we separate ITN topological properties that are the sheer outcome of randomness from those that are instead statistically significant? (4) Is the gravity model of trade able to replicate the observed ITN structure? (5) Can we explain the properties of the ITN in terms of standard economic forces such as country specialization and comparative advantage?