Geodesic currents are measures that realize a suitable closure of the space of multi-curves on a surface. They were introduced by Bonahon in 1986, when he proved that the Teichmueller space embeds inside the space of currents and, furthermore, hyperbolic length of curves can be extended to currents.
Since then, many other geometric structures on surfaces have been realized as currents and other functions on curves have been extended to currents, such as, recently, lengths coming from certain Anosov representations.
One of the key properties of these functions is that they decrease under surgery of an essential crossing of a curve, a phenomenon we refer to as the ``smoothing property''. In this talk we introduce the concept of geodesic current and discuss recent results which highlight the prominent role of the smoothing property. For example, functions on the space of curves that satisfy the smoothing property together with some other mild conditions, can be extended to geodesic currents continuously.