**Title:** Elliptic-elliptic surfaces

**Abstract:** Elliptic surfaces are a fairly well understood class of complex projective surfaces. They come with two discrete invariants $g$ and $d$, which are nonnegative integers. I will discuss some new results (joint with P. Engel, A. Ward, and Y. Zhang) about the moduli space and Hodge theory of elliptic surfaces with $(g,d)=(1,1)$. They behave like K3 surfaces in some respects, and they provide an interesting test case for the Hodge Conjecture in dimension 4.