A family of planar curves C_t is said to converge to a tropical curve C in R^2 if the corresponding family of amoebas A(C_t) converges in Hausdorff distance to C (after some rescaling). Our aim is to understand this convergence abstractly, in terms of the moduli of the underling family of Riemann surfaces. Doing so, we come to an abstract notion of tropical convergence of families in M_g to abstract tropical curves. This notion allows to keep track of the periods of the Riemann surfaces of the family, unifying two already existing concepts: the Jacobians of algebraic curves and the Jacobians of Tropical curves. This approach has potential applications in classical problems: compactification of moduli spaces, Riemann-Schottky, Brill-Noether. We will try to introduce every object carefully. In particular, no background in tropical geometry should be required. |

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