We prove that the (divisorial) gonality of a finite connected graph is lower bounded by its treewidth. Graphs for which equality holds include the grid graphs and the complete multipartite graphs. We prove that the treewidth lower bound also holds for metric graphs (tropical curves) by constructing for any positive rank divisor on a metric graph a positive rank divisor of the same degree on a subdivision of the underlying combinatorial 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.