  In this paper we construct various moduli spaces of K3 surfaces $M$ equipped
with a surjective holomorphic map $\pi:M\to\Pb^1$ with generic fiber a complex
torus (e.g., an elliptic fibration). Examples include moduli spaces of such
maps with primitive fibers; with reduced, irreducible fibers; equipped with a
section; etc. Such spaces are closely related to the moduli space of Ricci-flat
metrics on $M$. We construct period mappings relating these moduli spaces to
locally symmetric spaces, and use these to compute their (orbifold) fundamental
groups.
  These results lie in contrast to, and exhibit different behavior than, the
well-studied case of moduli spaces of polarized K3 surfaces, and are more
useful for applications to the mapping class group $\Mod(M)$. Indeed, we apply
our results on moduli space to give two applications to the smooth mapping
class group of $M$.
