  We consider the following game, played on a $k$-uniform hypergraph $H$. There
are $q$ colors available and two players take it in turns to color vertices. A
partial coloring is proper if no edge is mono-chromatic. One player, A, wishes
to color all the vertices and the other player, B, wishes to prevent this. The
{\em game chromatic number} $\chi_g(H)$ is the minimum number of colors for
which A has a winning strategy. We consider this in the context of a random
$k$-uniform hypergraph and prove upper and lower bounds that hold w.h.p.
