  We provide a proof, in $ZFC$, of Shelah's eventual categoricity conjecture
for abstract elementary classes (AEC's). Moreover, assuming in addition the
Singular Cardinal Hypothesis ($SCH$), we prove a direct generalization to the
more general context of accessible categories with directed colimits. If
$\mathcal{K}$ is such a category, we show that there is a cardinal $\mu$ such
that if $\mathcal{K}$ is $\lambda$-categorical for some $\lambda \geq \mu$
(i.e., it has only one object of internal size $\lambda$ up to isomorphism),
then $\mathcal{K}$ is eventually categorical (i.e., it is
$\lambda'$-categorical for every $\lambda' \geq \mu$). When considering
cardinalities of models of infinitary theories $\mathbb{T}$ of
$\mathcal{L}_{\kappa, \theta}$ that axiomatize $\mathcal{K}$, the result
implies, under $SCH$, the following infinitary version of Morley's categoricity
theorem: let $S$ be the class of cardinals $\lambda$ which are of cofinality at
least $\theta$ but are not successors of cardinals of cofinality less than
$\theta$. Then, if $\mathbb{T}$ is a $\mathcal{L}_{\kappa, \theta}$ theory
whose models have directed colimits and it is $\lambda$-categorical for some
$\lambda \geq \mu$ in $S$, then it is $\lambda'$-categorical for every
$\lambda' \geq \mu$ in $S$; moreover, we also exhibit an example that shows
that the exceptions in the class $S$ are needed. Along the way we also prove
Grossberg conjecture, according to which categoricity in a high enough cardinal
implies eventual amalgamation. We establish this result in AEC's and, assuming
in addition $SCH$, in the more general context of accessible categories whose
morphisms are monomorphisms.
