  We review and extend several recent results on the existence of the ground
state for the nonlinear Schr\"odinger (NLS) equation on a metric graph. By
ground state we mean a minimizer of the NLS energy functional constrained to
the manifold of fixed $L^2$-norm. In the energy functional we allow for the
presence of a potential term, of delta-interactions in the vertices of the
graph, and of a power-type focusing nonlinear term. We discuss both subcritical
and critical nonlinearity. Under general assumptions on the graph and the
potential, we prove that a ground state exists for sufficiently small mass,
whenever the constrained infimum of the quadratic part of the energy functional
is strictly negative.
