The Goldman bracket and the intersection of curves on surfaces

In this note, we discuss a Lie algebra structure of Goldman from an elementary point of view, together with its relation to the structure of intersection and self-intersection of curves on surfaces. We also list examples and mention some of the open problems in the area. This Lie algebra is defined by combining two well known operations on homotopy classes of curves: the transversal intersection and the composition of directed loops which start and end at the same point. The Lie algebra turns out to be a powerful tool and its structure still contains many mysteries.