  In this paper, we characterize the possible cofinalities of the least
$\lambda$-strongly compact cardinal. We show that, on the one hand, for any
regular cardinal, $\delta$, that carries a $\lambda$-complete uniform
ultrafilter, it is consistent, relative to the existence of a supercompact
cardinal above $\delta$, that the least $\lambda$-strongly compact cardinal has
cofinality $\delta$. On the other hand, provably the cofinality of the least
$\lambda$-strongly compact cardinal always carries a $\lambda$-complete uniform
ultrafilter.
