  In this paper we address a problem posed by Shelah in 1999 to find a suitable
notion for superstability for abstract elementary classes in which limit models
of cardinality $\mu$ are saturated.
  Theorem 1. Suppose that $\mathcal{K}$ is a $\chi$-tame abstract elementary
class with no maximal models satisfying the joint embedding property and the
amalgamation property. Suppose $\mu$ is a cardinal with
$\mu\geq\beth_{(2^{LS(\mathcal{K})+\chi})^+}$. Let $M$ be a model of
cardinality $\mu$. If $\mathcal{K}$ is both $\chi$-stable and $\mu$-stable and
satisfies the $\mu$-superstability assumptions, then any two $\mu$-limit models
over $M$ are isomorphic over $M$.
  Moreover, we identify sufficient conditions for superlimit models of
cardinality $\mu$ to exist, for model homogeneous models to be superlimit, and
for a union of saturated models to be saturated.
