  In this paper we study topology of moduli spaces of tropical curves of genus
$g$ with $n$ marked points. We view the moduli spaces as being imbedded in a
larger space, which we call the {\it moduli space of metric graphs with $n$
marked points.} We describe the shrinking bridges strong deformation
retraction, which leads to a substantial simplification of all these moduli
spaces. In the rest of the paper, that reduction is used to analyze the case of
genus 1. The corresponding moduli space is presented as a quotient space of a
torus with respect to the conjugation ${\mathbb Z}_2$-action; and furthermore,
as a homotopy colimit over a simple diagram. The latter allows us to compute
all Betti numbers of that moduli space with coefficients in ${\mathbb Z}_2$.
