  In this paper we study a multi-partite version of the Erd\H{o}s--Stone
theorem. Given integers $r<k$ and $t\ge 1$, let $ex_k(n, K_{r+1}(t))$ be the
maximum number of edges of $K_{r+1}(t)$-free $k$-partite graphs with $n$
vertices in each part, where $K_{r+1}(t)$ is the $t$-blowup of $K_{r+1}$. An
easy consequence of the supersaturaion result gives that $ex_k(n, K_{r+1}(t)) =
ex_k(n, K_{r+1})+o(n^2)$. Similar to a result of Erd\H os and Simonovits for
the non-partite case, we find that the error term is closely related to the
(multi-partite) Zarankiewicz problem. Using such Zarankiewicz numbers, for
$t=2,3$, we determine the error term up to an additive linear term; using some
natural assumptions on such Zarankiewicz numbers, we determine the error term
up to an additive constant depending on $k$, $r$ and $t$. We actually obtain
exact results in many cases, for example, when $k\equiv 0, 1 \pmod{r}$. Our
proof uses the stability method and starts by proving a stability result for
$K_{r+1}$-free multi-partite graphs.
