  Geodesic nets on Riemannian manifolds form a natural class of stationary
objects generalizing geodesics. Yet almost nothing is known about their
classification or general properties even when the ambient Riemannian manifold
is the Euclidean plane or the round $2$-sphere.
  In the first half of this paper we survey some results and open questions
(old and new) about geodesic nets on Riemannian manifolds. Many of these open
questions are about geodesic nets on the Euclidean plane. The second half
contains a partial answer for one of these questions, namely, a description of
a new infinite family of geodesic nets on the Euclidean plane with 14 boundary
(or unbalanced) vertices and arbitrarily many inner (or balanced) vertices of
degree $\geq 3$.
