  In this paper, we establish limit theorems for the supremum of the support,
denoted by $M_t$, of a supercritical super-Brownian motion $\{X_t, t\ge0\}$ on
$\mathbb{R}$. We prove that there exists an $m(t)$ such that $(X_t-m(t),
M_t-m(t))$ converges in law, and give some large deviation results for $M_t$ as
$t\to\infty$. We also prove that the limit of the extremal process
$\mathcal{E}_t:=X_t-m(t)$ is a Poisson random measure with exponential
intensity in which each atom is decorated by an independent copy of an
auxiliary measure. These results are analogues of the results for branching
Brownian motions obtained in Arguin et al. (Probab. Theory Relat. Fields 157
(2013), 535-574), A\"id\'ekon et al. (Probab. Theory Relat. Fields 157 (2013),
405-451) and Roberts (Ann. Probab. 41 (2013), 3518-3541).

Probability