In the first part of the talk we will discuss different aspects of the construction of moduli spaces of tropical curves, tropical abelian varieties along with the Torelli map among them. Moreover, we will present tropical versions of Teichmuller and Siegel spaces along with a lifting of the Torelli map. In the second part we will explain how these spaces can be seen as tropicalizations of the classical spaces. We will be focusing in particular on open problems related to these constructions.