  We study the phase diagram of the ferromagnetic $q$-state Potts model on the
various three-dimensional lattices for integer and non-integer values of $q>1$.
Our approach is based on a thermodynamically self-consistent Ornstein-Zernike
approximation for the two-point correlation functions. We calculate the
transition temperatures and, when the transition is first order, the jump
discontinuities in the magnetization and the internal energy, as well as the
coordinates of the critical endpoint in external field. Our predictions are in
very good agreement with best available estimates. From the numerical study of
the region of weak first-order transition, we estimate the critical value $q_c$
for which the transition changes from second to first-order. The $q->1$ limit
that describes the bond-percolation problem is also investigated.
