In this thesis we study the moduli space of genus g curves, and the differential forms that occur naturally on this moduli space. We show that the rings of these tautological differential forms are... Show moreIn this thesis we study the moduli space of genus g curves, and the differential forms that occur naturally on this moduli space. We show that the rings of these tautological differential forms are finite-dimensional, and discuss algorithms that can be used to compute relations among tautological differential forms. Show less