Modular varieties, by their very definition as parameter spaces,
usually carry very interesting cycles and enjoy intriguing arithmetic and
geometric properties. These lectures will concentrate on the birational
geometry of the moduli space of curves, as well as some of its modular
covers (Prym and spin moduli spaces together with the higher level
analogues). Topics include Kodaira dimension, singularities of moduli
spaces, and techniques for showing unirationality or rational
connectedness of these spaces in small genus.