Let K be a complete non-Archimedean field, let X be a smooth, complete, connected algebraic K-curve with split semistable reduction, and let J(X) be its Jacobian. Berkovich has defined a notion of the skeleton of X, which is a metric graph \Sigma(X) onto which the analytification X^an deformation retracts, as well as a notion of the skeleton of J(X), which is a real torus \Sigma(J(X)) onto which J(X)^an deformation retracts. Following earlier work of Baker, we develop a theory of the Jacobian J(\Gamma) of a metric graph \Gamma, which is again a real torus, and we show that \Sigma(J(X)) = J(\Sigma(X)) canonically. If K is discretely valued then this gives a method of calculating the component group of the Neron model of J(X) without appealing to Raynaud's theorem, and if K is algebraically closed then this gives a powerful tool for constructing meromorphic functions on X with prescribed absolute value. This work is joint with Matt Baker.