  We prove:
  Main Theorem: Let $\mathcal{K}$ be an abstract elementary class satisfying
the joint embedding and the amalgamation properties with no maximal models of
cardinality $\mu$. Let $\mu$ be a cardinal above the the L\"owenheim-Skolem
number of the class. If $\mathcal{K}$ is $\mu$-Galois-stable, has no
$\mu$-Vaughtian Pairs, does not have long splitting chains, and satisfies
locality of splitting, then any two $(\mu,\sigma_\ell)$-limits over $M$, for
$\ell\in\{1,2\}$, are isomorphic over $M$.
  This theorem extends results of Shelah from \cite{Sh394}, \cite{Sh576},
\cite{Sh600}, Kolman and Shelah in \cite{KoSh} and Shelah and Villaveces from
\cite{ShVi}. A preliminary version of our uniqueness theorem, which was
circulated in 2006, was used by Grossberg and VanDieren to prove a case of
Shelah's categoricity conjecture for tame abstract elementary classes in
\cite{GrVa2}. Preprints of this paper have also influenced the Ph.D. theses of
Drueck \cite{Dr} and Zambrano \cite{Za}. This paper also serves the expository
role of presenting together the arguments in \cite{Va1} and \cite{Va2} in a
more natural context in which the amalgamation property holds and this work
provides an approach to the uniqueness of limit models that does not rely on
Ehrenfeucht-Mostowski constructions.
