A geodesic net is an embedded graph in a Riemannian manifold that is a stationary point of the length functional on the space of embedded graphs. It is a natural generalization of a closed geodesic.

I will give an introduction to the theory of geodesic nets and report on joint work with Gorodkov, Frolov, and Nabutovsky, in which we prove that the number of branching points of a geodesic net on a Riemannian surface can be bounded solely in terms of the geometry of the surface and the length of the net. In particular, it follows from the estimate that for sufficiently large $L$, the number of branching points is at most $(400L{)}^{(180L{)}^4}$.