  We study the static as well as the glassy or dynamical transition in the
mean-field $p$-state Potts glass. By numerical solution of the saddle point
equations we investigate the static and the dynamical transition for all values
of $p$ in the non-perturbative regime $p>4$. The static and dynamical
Edwards-Anderson parameter increase with $p$ logarithmically. This makes the
glassy transition temperature lie very close to the static one. We compare the
main predictions of the theory with the numerical simulations.
