We introduce the basics on geodesic currents on surfaces, survey a few recent results in relation to them, and discuss further generalizations in the setting of hyperbolic groups.

Geodesic currents are measures supported on subsets of geodesics of a surface which can be seen as a suitable closure of the space of all closed geodesics. In fact, they encode a lot of geometric information of the surface: many geometric structures on the surface embed inside the space of geodesic currents, such as Teichmueller space. The space of measures is locally compact, has a pairing that extends the notion of intersection number on curves, and is equipped with a very natural metric. In this mini-course we will introduce some of these features, and study currents from the equivalent perspectives of pseudo-metrics on the surface and as well as their associated marked length spectrum.