  We investigate the existence of ground states with fixed mass for the
nonlinear Schr\"odinger equation with a pure power nonlinearity on periodic
metric graphs. Within a variational framework, both the $L^2$-subcritical and
critical regimes are studied. In the former case, we establish the existence of
global minimizers of the NLS energy for every mass and every periodic graph. In
the critical regime, a complete topological characterization is derived,
providing conditions which allow or prevent ground states of a certain mass
from existing. Besides, a rigorous notion of periodic graph is introduced and
discussed.
