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We compactify the classical moduli variety of compact Riemann surfaces byattaching moduli of (metrized) graphs as boundary. The compactifications do notadmit the structure of varieties and patch together to form a big connectedmoduli space in which $\sqcup_{g} M_{g}$ is open dense.  The metrized graphs, which are often studied as "tropical curves", areobtained as Gromov-Hausdorff collapse by fixing diameters of the hyperbolicmetrics of the Riemann surfaces. This phenomenon can be also seen as anarchemidean analogue of the tropicalization of Berkovich analytification of$M_{g}$ (cf., [ACP]).

Tropical Geometric Compactification of Moduli, I - $M_g$ case -