  We show that Shelah's Eventual Categoricity Conjecture follows from the
existence of class many strongly compact cardinals. This is the first time the
consistency of this conjecture has been proven. We do so by showing that every
AEC with $LS(K)$ below a strongly compact cardinal $\kappa$ is $< \kappa$ tame
and applying the categoricity transfer of Grossberg and VanDieren. These
techniques also apply to measurable and weakly compact cardinals and we prove
similar tameness results under those hypotheses. We isolate a dual property to
tameness, called \emph{type shortness}, and show that it follows similarly from
large cardinals.
