  The Curie-Weiss Potts model is a mean field version of the well-known Potts
model. In this model, the critical line $\beta = \beta_c (h)$ is explicitly
known and corresponds to a first order transition when $q > 2$. In the present
paper we describe the fluctuations of the density vector in the whole domain
$\beta \geqslant 0$ and $h \geqslant 0$, including the conditional fluctuations
on the critical line and the non-Gaussian fluctuations at the extremity of the
critical line. The probabilities of each of the two thermodynamically stable
states on the critical line are also computed. Similar results are inferred for
the Random-Cluster model on the complete graph.
