We show that a tilted algebra $A$ is tame if and only if for each generic
root $\dd$ of $A$ and each indecomposable irreducible component $C$ of
$\module(A,\dd)$, the field of rational invariants $k(C)^{\GL(\dd)}$ is
isomorphic to $k$ or $k(x)$. Next, we show that the tame tilted algebras are
precisely those tilted algebras $A$ with the property that for each generic
root $\dd$ of $A$ and each indecomposable irreducible component $C \subseteq
\module(A,\dd)$, the moduli space $\M(C)^{ss}_{\theta}$ is either a point or
just $\mathbb P^1$ whenever $\theta$ is an integral weight for which
$C^s_{\theta}\neq \emptyset$. We furthermore show that the tameness of a tilted
algebra is equivalent to the moduli space $\M(C)^{ss}_{\theta}$ being smooth
for each generic root $\dd$ of $A$, each indecomposable irreducible component
$C \subseteq \module(A,\dd)$, and each integral weight $\theta$ for which
$C^s_{\theta} \neq \emptyset$. As a consequence of this latter description, we
show that the smoothness of the various moduli spaces of modules for a strongly
simply connected algebra $A$ implies the tameness of $A$.
Along the way, we explain how moduli spaces of modules for finite-dimensional
algebras behave with respect to tilting functors, and to theta-stable
decompositions.