  In this paper we establish some conditional limit theorems for some critical
superprocesses $X=\{X_t, t\ge 0\}$. First we identify the rate of
non-extinction. Then we show that, for a large class of functions $f$,
conditioned on non-extinction at time $t$, the limit, as $t\to\infty$, of
$t^{-1}\langle f, X_t\rangle$ exists in distribution and we identify this
limit. Finally, we also establish, under some conditions, a central limit
theorem for $\langle f, X_t\rangle$ conditioned on non-extinction at time $t$.
