| In a series of well-known papers from the nineties, Mukai dicovered new models for the moduli space M_g of curves of genus g<12, by using moduli of K3 surfaces. In joint work with A. Verra, we present a striking analogy of this type of result at the level of the Prym moduli space R_g and many of Mukai's results for M_g have precise Prym counterparts. Via K3 surfaces of Nikulin type, we show that R_g has a Nikulin-Mukai model for g<8 (which establishes the unirationality of R_g in this range), while R_7 has the structure of a Mori fibre space over the moduli space of Nikulin surfaces. We finally establish the uniruledness of R_8, which is the highest genus where such a result is known to hold. |