Let $T$ be a general Enriques surface and let $f:S\to T$ be its universal cover. Consider a smooth curve $C\subset T$ of genus $g\ge 2$, set $D:=f-1(C)$, and let $\backslash mcC\to |C|$ and $\backslash mcD\to f\ast |C|\subset |D|$ be the universal family and the restriction of the universal family respectively. We construct the relative Prym variety $P=\backslash Prym(\backslash mcD,\backslash mcC)$ of $\backslash mcD$ over $\backslash mcC$ and show that it is a (possibly singular) symplectic variety of dimension $2g-2$. There is a morphism $P\to |C|$, which is a Lagrangian fibration and whose smooth fibers are $(g-1)$-dimensional Prym varieties. We also prove that the smooth locus of $P$ is simply connected. For any non zero integer $\chi =d-g+1$ we consider the degree $d$ relative compactified Jacobian $N=\backslash JacAd(|C|)\to |C|$, with respect to a polarization $A$ on $T$. If $\chi$ is such that the Mukai vector $(0,[D],2\chi )$ is primitive in $H\ast (S,Z)$, and $A$ if is general, we prove that $N$ is smooth. Moreover, under some technical assumption that can be verified for low values of $g$ and that are expected to be true in general, we show that $\pi 1(N)\cong \backslash Z/(2)$, that $\omega N\cong \backslash mcON$, and that $hp,0(N)=0$ for $p\ne 0,2g-1$.

The tautological ring of the moduli space of curves M_g is a subring R^(M_g) of the Chow ring A^(M_g). The tautological ring can also be defined for other moduli spaces of curves, such as the moduli space of curves of compact type M^c_g or the moduli space of Deligne-Mumford stable pointed curves Mbar_{g,n}. We conjecture and prove various results about the structure of the tautological ring.

In particular, we give two proofs of the Faber-Zagier relations, a large family of relations between the kappa classes in R^*(M_g) that contains all known relations. The first proof (joint work with R. Pandharipande) uses the virtual geometry of the moduli space of stable quotients developed by Marian, Oprea, and Pandharipande. The second proof (joint work with R. Pandharipande and D. Zvonkine) uses Witten's class on the moduli space of 3-spin curves and the classification of semisimple cohomological field theories by Givental and Teleman. The second proof has the disadvantage that it only proves the image of the Faber-Zagier relations in cohomology, but the advantage that it also proves an extension of the relations to Mbar_{g,n} that was conjectured by the author. These relations on Mbar_{g,n} and their restrictions to smaller moduli spaces of curves seem to describe all known relations in the tautological ring.

We also prove several combinatorial results about the structure of the Gorenstein quotient rings of R^(M_g) and R^(M^c_g). This includes several new families of relations that are similar to the Faber-Zagier relations, as well as joint work with F. Janda giving formulas for ranks of restricted socle pairings in R^*(M^c_g).

The appendix presents data obtained by computer calculations of the tautological relations on Mbar_{g,n} and their restrictions to M^c_{g,n} and M^{rt}_{g,n} for small values of g and n. The data suggests several new locations in which the tautological ring might not be a Gorenstein ring.