  Five-dimensional $\mathcal{N}=1$ theories with gauge group $U(N)$, $SU(N)$,
$USp(2N)$ and $SO(N)$ are studied at large rank through localization on a large
sphere. The phase diagram of theories with fundamental hypermultiplets is
universal and characterized by third order phase transitions, with the
exception of $U(N)$, that shows both second and third order transitions. The
phase diagram of theories with adjoint or (anti-)symmetric hypermultiplets is
also determined and found to be universal. Moreover, Wilson loops in
fundamental and antisymmetric representations of any rank are analyzed in this
limit. Quiver theories are discussed as well. All the results substantiate the
$\mathcal{F}$-theorem.
