  We study the moduli space of metric graphs that arise from tropical plane
curves. There are far fewer such graphs than tropicalizations of classical
plane curves. For fixed genus $g$, our moduli space is a stacky fan whose cones
are indexed by regular unimodular triangulations of Newton polygons with $g$
interior lattice points. It has dimension $2g+1$ unless $g \leq 3$ or $g = 7$.
We compute these spaces explicitly for $g \leq 5$.
