In this talk I will recall a few basic notions in the theory of currents as well as in tropical geometry. I will also introduce the notion of ``complex tropical currents'' which are closed normal currents on (C*)^n associated to tropical cycles in R^n. I will explain how to read the extremality properties of these tropical currents from the combinatorial data of the underlying tropical cycles and how to obtain an intersection theory for the tropical cycles from an intersection theory of currents. The main results to be mentioned in talk can be found at arXiv:1403.7456.