Moduli spaces of K3 surfaces are fundamental objects in algebraic geometry. Elliptic K3 surfaces are K3 surfaces with elliptic fibration structure, and they are of particular interest due to their rich geometry. The moduli space of elliptic K3 surfaces can be studied using the theory of Weierstrass models. In this dissertation, we study the topology and intersection theory of the moduli space of elliptic K3 surfaces.

We compute the Poincar e polynomial of the moduli space of elliptic K3 surfaces. The main idea is constructing a compactification using the Weierstrass models, this compactification is a GIT quotient. We adapt Kirwan's blowup machinery to weighted projective space to compute the Poincar e polynomial. We find the cohomology is mostly concentrated in the even degrees, but there is one odd degree class in degree $33$.

We also study the Chow ring of the moduli space of elliptic surfaces of degree $N\geq 2$. We conclude that the Chow ring of the moduli space of elliptic surfaces is always generated by two classes. Furthermore, explicit relations between these classes are given, the Poincar e polynomial for the Chow ring is the same for any $N\geq 2$ and the ring is Gorenstein with socle in degree $16$. When $N=2$, we obtain the Chow ring for the moduli space of elliptic K3 surfaces, we conclude that the Chow ring in this case is tautological.

Finally, we present localization computations on the relative Quot scheme over the moduli space of elliptic K3 surfaces. Our calculations are sufficient to determine the divisorial $\kappa$-classes in terms of the Hodge class. We also represent one Noether-Lefschetz divisor in terms of the Hodge class, which agrees with the modularity nature of the Noether-Lefschetz divisors.