We give explicit descriptions of the moduli spaces of abelian surfaces with polarizations of type (1, d), for d = 12, 14, 16, 18 and 20. More precisely, in each case we show that a certain choice of moduli space of such abelian surfaces with a partial level structure can be described explicitly and is unirational, and in some cases rational. These moduli spaces with partial level structure are covers of the ordinary moduli spaces, so the Kodaira dimension of the ordinary moduli spaces in these

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... ases is −∞. In addition, we give a few new examples of Calabi-Yau three-folds fibred in abelian surfaces. In the case of d = 20, such Calabi-Yau three-folds play a key role in the description of the abelian surfaces.