  In this paper, we determine the achromatic and diachromatic numbers of some
circulant graphs and digraphs each one with two lengths and give bounds for
other circulant graphs and digraphs with two lengths. In particular, for the
achromatic number we state that $\alpha(C_{16q^2+20q+7}(1,2))=8q+5$, and for
the diachromatic number we state that
$dac(\overrightarrow{C}_{32q^2+24q+5}(1,2))=8q+3$. In general, we give the
lower bounds $\alpha(C_{4q^2+aq+1}(1,a))\geq 4q+1$ and
$dac(\overrightarrow{C}_{8q^2+2(a+4)q+a+3}(1,a))\geq 4q+3$ when $a$ is a non
quadratic residue of $\mathbb{Z}_{4q+1}$ for graphs and $\mathbb{Z}_{4q+3}$ for
digraphs, and the equality is attained, in both cases, for $a=3$. Finally, we
determine the achromatic index for circulant graphs of $q^2+q+1$ vertices when
the projective cyclic plane of odd order $q$ exists.
