Summary: A metrized complex of algebraic curves over an algebraically closed field \(\kappa\) is, roughly speaking, a finite metric graph \(\Gamma\) together with a collection of marked complete nonsingular algebraic curves \(C_v\) over \(\kappa\), one for each vertex \(v\) of \(\Gamma\); the marked points on \(C_v\) are in bijection with the edges of \(\Gamma\) incident to \(v\). We define linear equivalence of divisors and establish a Riemann-Roch theorem for metrized complexes of curves which combines the classical Riemann-Roch theorem over \(\kappa\) with its graph-theoretic and tropical analogues from the first author and L. Caporaso [Adv. Math. 240, 1–23 (2013; Zbl 1284.14087)]; the second author and S. Norine [Adv. Math. 215, No. 2, 766–788 (2007; Zbl 1124.05049)]; A. Gathmann and M. Kerber [Math. Z. 259, No. 1, 217–230 (2008; Zbl 1187.14066)] and G. Mikhalkin and I. Zharkov [Contemp. Math. 465, 203–230 (2008; Zbl 1152.14028)], providing a common generalization of all of these results. For a complete nonsingular curve \(X\) defined over a non-Archimedean field \(\mathbb K\), together with a strongly semistable model \(\mathfrak X\) for \(X\) over the valuation ring \(R\) of \(\mathbb K\), we define a corresponding metrized complex \(\mathfrak{CX}\) of curves over the residue field \(\kappa\) of \(\mathbb K\) and a canonical specialization map \(\tau^{\mathfrak {CX}}_\ast\) from divisors on \(X\) to divisors on \(\mathfrak{CX}\) which preserves degrees and linear equivalence. We then establish generalizations of the specialization lemma from the second author [Algebra Number Theory 2, No. 6, 613–653 (2008; Zbl 1162.14018)] and its weighted graph analogue from O. Amini and L. Caporaso [Adv. Math. 240, 1–23 (2013; Zbl 1284.14087)], showing that the rank of a divisor cannot go down under specialization from \(X\) to \(\mathfrak{CX}\). As an application, we establish a concrete link between specialization of divisors from curves to metrized complexes and the theory of limit linear series due to D. Eisenbud and J. Harris [Invent. Math. 85, 337–371 (1986; Zbl 0598.14003)]. Using this link, we formulate a generalization of the notion of limit linear series to curves which are not necessarily of compact type and prove, among other things, that any degeneration of a \(\mathfrak g^r_d\) in a regular family of semistable curves is a limit \(\mathfrak g^r_d\) on the special fiber.