  We prove that the (divisorial) gonality of a finite connected graph is lower
bounded by its treewidth. We show that equality holds for grid graphs and
complete multipartite graphs.
  We prove that the treewidth lower bound also holds for \emph{metric graphs}
by constructing for any positive rank divisor on a metric graph $\Gamma$ a
positive rank divisor of the same degree on a subdivision of the underlying
graph.
  Finally, we show that the treewidth lower bound also holds for a related
notion of gonality defined by Caporaso and for stable gonality as introduced by
Cornelissen et al.
