  We consider compactness characterizations of large cardinals. Based on
results of Benda \cite{b-sccomp}, we study compactness for omitting types in
various logics. In $\bL_{\kappa, \kappa}$, this allows us to characterize any
large cardinal defined in terms of normal ultrafilters, and we also analyze
second-order and sort logic. In particular, we give a compactness for omitting
types characterization of huge cardinals, which have consistency strength
beyond Vop\v{e}nka's Principle.
