  In this paper, we study the structure, deformations and the moduli spaces of
complex projective surfaces admitting genus two fibrations over elliptic
curves. We observe that, a surface admitting a smooth fibration as above is
elliptic and we employ results on the moduli of polarized elliptic surfaces, to
construct moduli spaces of these smooth fibrations. In the case of nonsmooth
fibrations, we relate the moduli spaces to the Hurwitz schemes ${\cal
H}(1,X(d),n)$ of morphisms of degree $n$ from elliptic curves to the modular
curve $X(d)$, $d\geq 3$. Ultimately, we show that the moduli spaces in the
nonsmooth case are fiber spaces over the affine line ${\mathbb A}^1$ with
fibers determined by the components of ${\cal H}(1,X(d),n)$.
