  In the present paper an introduction to the new subject of nonlinear
dispersive hamiltonian equations on graphs is given. The focus is on recently
established properties of solutions in the case of nonlinear Schr\"odinger
equation. Special consideration is given to existence and behaviour of solitary
solutions. Two subjects are discussed in some detail concerning NLS equation on
a star graph: the standing waves of NLS equation on a graph with a $\delta$
interaction at the vertex; the scattering of fast solitons through an
Y-junction in the cubic case. The emphasis is on description of concepts and
results and on physical context, without reporting detailed proofs; some
perspectives and more ambitious open problems are discussed.
