With one partial exception, the birational type of the moduli space of principally polarised complex abelian g-folds is now known, and there are also results for related spaces such as Prym moduli spaces, universal families, non-principal polarisation and others. In general such spaces are of general type for large values of the parameters (g, degree etc.) and at least uniruled for small values. A few of the moduli spaces are known to be rational. In this talk I shall start to address questions of rationality and unirationality for these spaces over non-closed fields, usually the field of rational numbers. Little seems to be known but I hope to give some answers in simple cases, and to describe some consequences that follow or would follow from such results.