  In this paper is studied ferromagnetic three states Potts model on a Cayley
tree of order three and we give explicit formulas for translation-invariant
Gibbs measures. Furthermore, we show that under some conditions on the
parameter of the antiferromagnetic Potts model with q-states with zero external
field on the Cayley tree of order $k>2$, there are exactly 2(2^q-1) periodic
(non translation-invariant) Gibbs measures.
