  We consider the $t$-improper chromatic number of the Erd{\H o}s-R{\'e}nyi
random graph $G(n,p)$. The t-improper chromatic number $\chi^t(G)$ of $G$ is
the smallest number of colours needed in a colouring of the vertices in which
each colour class induces a subgraph of maximum degree at most $t$. If $t = 0$,
then this is the usual notion of proper colouring. When the edge probability
$p$ is constant, we provide a detailed description of the asymptotic behaviour
of $\chi^t(G(n,p))$ over the range of choices for the growth of $t = t(n)$.
